{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:53:50Z","timestamp":1776844430919,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"240","license":[{"start":{"date-parts":[[2002,12,5]],"date-time":"2002-12-05T00:00:00Z","timestamp":1039046400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We numerically compute the central critical values of odd quadratic character twists with respect to some small discriminants\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper D\">\n                        <mml:semantics>\n                          <mml:mi>D<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">D<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of spinor zeta functions attached to Seigel\u2013Hecke eigenforms\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F\">\n                        <mml:semantics>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of genus 2 in the first few cases where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F\">\n                        <mml:semantics>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    does not belong to the Maass space. As a result, in the cases considered we can numerically confirm a conjecture of B\u00f6cherer, according to which these central critical values should be proportional to the squares of certain finite sums of Fourier coefficients of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F\">\n                        <mml:semantics>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01399-0","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T14:12:48Z","timestamp":1032531168000},"page":"1597-1607","source":"Crossref","is-referenced-by-count":10,"title":["Some numerical computations concerning spinor zeta functions in genus 2 at the central point"],"prefix":"10.1090","volume":"71","author":[{"given":"Winfried","family":"Kohnen","sequence":"first","affiliation":[]},{"given":"Michael","family":"Ku\u00df","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,12,5]]},"reference":[{"key":"1","series-title":"A Wiley-Interscience Publication","isbn-type":"print","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0471800074"},{"issue":"3","key":"2","first-page":"43","article-title":"Euler products that correspond to Siegel\u2019s modular forms of genus 2","volume":"29","author":"Andrianov, A. N.","year":"1974","journal-title":"Uspehi Mat. Nauk","ISSN":"https:\/\/id.crossref.org\/issn\/0042-1316","issn-type":"print"},{"key":"3","unstructured":"[3] S. B\u00f6cherer: Bemerkungen \u00fcber die Dirichletreihen von Koecher und Maass. Math. Gottingensis, Schriftenr. d. Sonderforschungsbereichs Geom. Anal. 68, (1986)."},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/BF02570834","article-title":"The Dirichlet series of Koecher and Maass and modular forms of weight \\frac32","volume":"209","author":"B\u00f6cherer, Siegfried","year":"1992","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"5","series-title":"Progress in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9162-3","volume-title":"The theory of Jacobi forms","volume":"55","author":"Eichler, Martin","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0817631801"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"103","DOI":"10.2206\/kyushumfs.47.103","article-title":"On characteristic twists of certain Dirichlet series","volume":"47","author":"Kohnen, Winfried","year":"1993","journal-title":"Mem. Fac. Sci. Kyushu Univ. Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0373-6385","issn-type":"print"},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1007\/BF02570489","article-title":"Characteristic twists of a Dirichlet series for Siegel cusp forms","volume":"87","author":"Kohnen, W.","year":"1995","journal-title":"Manuscripta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-2611","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1007\/BF01393889","article-title":"A certain Dirichlet series attached to Siegel modular forms of degree two","volume":"95","author":"Kohnen, W.","year":"1989","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"9","unstructured":"[9] M. Ku\u00df: Die getwistete Spinor Zeta Funktion und die B\u00f6cherer Vermutung. Dissertation. (2000)"},{"key":"10","first-page":"278","article-title":"Functional equations of the Dirichlet functions","volume":"171","author":"Lavrik, A. F.","year":"1966","journal-title":"Dokl. Akad. Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"key":"11","doi-asserted-by":"publisher","first-page":"521","DOI":"10.2307\/1968936","article-title":"A class of orthogonal functions on plane curves","volume":"40","author":"Jackson, Dunham","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"197","key":"12","doi-asserted-by":"publisher","first-page":"381","DOI":"10.2307\/2153042","article-title":"Computations of Siegel modular forms of genus two","volume":"58","author":"Skoruppa, Nils-Peter","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"13","first-page":"375","article-title":"Sur les coefficients de Fourier des formes modulaires de poids demi-entier","volume":"60","author":"Waldspurger, J.-L.","year":"1981","journal-title":"J. Math. Pures Appl. (9)","ISSN":"https:\/\/id.crossref.org\/issn\/0021-7824","issn-type":"print"},{"key":"14","unstructured":"[14] R. Weissauer: The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula). Preprint, Mannheim (1993)"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01399-0\/S0025-5718-01-01399-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01399-0\/S0025-5718-01-01399-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:04:13Z","timestamp":1776726253000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01399-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,12,5]]},"references-count":14,"journal-issue":{"issue":"240","published-print":{"date-parts":[[2002,10]]}},"alternative-id":["S0025-5718-01-01399-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01399-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2001,12,5]]}}}