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Using Weil\u2019s explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known discriminant.<\/p>","DOI":"10.1090\/s0025-5718-01-01305-9","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1607-1616","source":"Crossref","is-referenced-by-count":3,"title":["Localization of the first zero of the Dedekind zeta function"],"prefix":"10.1090","volume":"70","author":[{"given":"Sami","family":"Omar","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,3,7]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"[CDO] H. Cohen, F. Diaz y Diaz and M. Olivier, A table of totally complex number fields of small discriminants, Algorithmic Number Theory, (Lectures Notes in Computer Science; Springer Verlag, 1998), pp. 381\u2013391.","DOI":"10.1007\/BFb0054877"},{"key":"2","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02945-9","volume-title":"A course in computational algebraic number theory","volume":"138","author":"Cohen, Henri","year":"1993","ISBN":"http:\/\/id.crossref.org\/isbn\/3540556400"},{"key":"3","series-title":"Computer Science and Applied Mathematics","volume-title":"Methods of numerical integration","author":"Davis, Philip J.","year":"1975"},{"key":"4","series-title":"Publications de l'Institut de Math\\'{e}matique de l'Universit\\'{e} de Nancago, No. IX","volume-title":"Les nombres premiers","author":"Ellison, William John","year":"1975"},{"key":"5","first-page":"Exp. No. 5, 23","article-title":"Hecke\u2019s integral formula","author":"Friedman, Eduardo","year":"[1988?]"},{"key":"6","isbn-type":"print","volume-title":"Table of integrals, series, and products","author":"Gradshteyn, I. S.","year":"1980","ISBN":"http:\/\/id.crossref.org\/isbn\/0122947606"},{"key":"7","volume-title":"Algebraic number theory","author":"Lang, Serge","year":"1970"},{"key":"8","volume-title":"Vorlesungen \\\"{u}ber Zahlentheorie. Erster Band, zweiter Teil; zweiter Band; dritter Band","author":"Landau, Edmund","year":"1969"},{"issue":"1","key":"9","doi-asserted-by":"crossref","first-page":"119","DOI":"10.5802\/jtnb.22","article-title":"Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results","volume":"2","author":"Odlyzko, A. M.","year":"1990","journal-title":"S\\'{e}m. Th\\'{e}or. Nombres Bordeaux (2)","ISSN":"http:\/\/id.crossref.org\/issn\/0989-5558","issn-type":"print"},{"key":"10","first-page":"235","article-title":"The minimum discriminant of seventh degree totally real algebraic number fields","author":"Pohst, M.","year":"1977"},{"key":"11","unstructured":"[Pt1] G. Poitou, Minorations de discriminants, S\u00e9minaire Bourbaki, 28e ann\u00e9e, 1975\/76, n 479."},{"key":"12","first-page":"136","article-title":"Minorations de discriminants (d\u2019apr\u00e8s A. M. Odlyzko)","author":"Poitou, Georges","year":"1977"},{"issue":"219","key":"13","doi-asserted-by":"publisher","first-page":"1295","DOI":"10.1090\/S0025-5718-97-00871-5","article-title":"Zeros of Dedekind zeta functions in the critical strip","volume":"66","author":"Tollis, Emmanuel","year":"1997","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01305-9\/S0025-5718-01-01305-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01305-9\/S0025-5718-01-01305-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:38:18Z","timestamp":1627605498000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01305-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,3,7]]},"references-count":13,"journal-issue":{"issue":"236","published-print":{"date-parts":[[2001,10]]}},"alternative-id":["S0025-5718-01-01305-9"],"URL":"http:\/\/dx.doi.org\/10.1090\/s0025-5718-01-01305-9","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":["Applied Mathematics","Computational Mathematics","Algebra and Number Theory"],"published":{"date-parts":[[2001,3,7]]}}}