{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:19:43Z","timestamp":1776788383801,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"264","license":[{"start":{"date-parts":[[2009,6,2]],"date-time":"2009-06-02T00:00:00Z","timestamp":1243900800000},"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":"

\n Let\n \n \n \n \n \n F<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n F<\/mml:mi>\n <\/mml:mrow>\n F\u2019\/F<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a finite normal extension of number fields with Galois group\n \n \n \n \n G<\/mml:mi>\n a<\/mml:mi>\n l<\/mml:mi>\n (<\/mml:mo>\n \n F<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Gal(F\u2019\/F)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let\n \n \n \n \n \u03c7\n \n <\/mml:mi>\n \\chi<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be an irreducible character of\n \n \n \n \n G<\/mml:mi>\n a<\/mml:mi>\n l<\/mml:mi>\n (<\/mml:mo>\n \n F<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Gal(F\u2019\/F)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of degree greater than one and\n \n \n \n \n L<\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n ,<\/mml:mo>\n \n \u03c7\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n L(s,\\chi )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the associated Artin\n \n \n \n L<\/mml:mi>\n L<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -function. Assuming the truth of Artin\u2019s conjecture, we have explicitly determined a zero-free region about\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n L<\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n ,<\/mml:mo>\n \n \u03c7\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n L(s,\\chi )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . As an application we show that, for a CM-field\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of degree\n \n \n \n \n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:mrow>\n 2n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with solvable normal closure over\n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , if\n \n \n \n \n n<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 370<\/mml:mn>\n <\/mml:mrow>\n n \\geq 370<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as well as\n \n \n \n \n n<\/mml:mi>\n \n \u2209\n \n <\/mml:mo>\n {<\/mml:mo>\n 384<\/mml:mn>\n ,<\/mml:mo>\n 400<\/mml:mn>\n ,<\/mml:mo>\n 416<\/mml:mn>\n ,<\/mml:mo>\n 448<\/mml:mn>\n ,<\/mml:mo>\n 512<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n n \\notin \\{ 384, 400, 416, 448, 512 \\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then the relative class number of\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is greater than one.\n <\/p>","DOI":"10.1090\/s0025-5718-08-02093-0","type":"journal-article","created":{"date-parts":[[2008,7,25]],"date-time":"2008-07-25T12:55:42Z","timestamp":1216990542000},"page":"2437-2445","source":"Crossref","is-referenced-by-count":0,"title":["The zeros of Dedekind zeta functions and class numbers of CM-fields"],"prefix":"10.1090","volume":"77","author":[{"given":"Geon-No","family":"Lee","sequence":"first","affiliation":[]},{"given":"Soun-Hi","family":"Kwon","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,6,2]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"213","DOI":"10.4064\/aa106-3-2","article-title":"Bounds for the degrees of CM-fields of class number one","volume":"106","author":"Bessassi, Sofi\u00e8ne","year":"2003","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"6","key":"2","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1155\/S1073792895000225","article-title":"Siegel zeros and cusp forms","author":"Hoffstein, Jeffrey","year":"1995","journal-title":"Internat. 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