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ONE-LEVEL DENSITY OF LOW-LYING ZEROS OF QUADRATIC HECKE L-FUNCTIONS OF IMAGINARY QUADRATIC NUMBER FIELDS

Part of: Zeta and $L$-functions: analytic theory Algebraic number theory: global fields Exponential sums and character sums

Published online by Cambridge University Press:  29 October 2020

PENG GAO  and
LIANGYI ZHAO Open the ORCID record for LIANGYI ZHAO [Opens in a new window]
PENG GAO
Affiliation:
School of Mathematical Sciences, Beihang University, Beijing100191, China e-mail: [email protected]
LIANGYI ZHAO*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW2052, Australia
*
e-mail: [email protected]
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Abstract

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

Keywords

one-level densitylow-lying zerosquadratic Hecke characterHecke L-functions

MSC classification

Primary: 11L40: Estimates on character sums
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 11M20: Real zeros of $L(s, chi)$; results on $L(1, chi)$ 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M50: Relations with random matrices 11R16: Cubic and quartic extensions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 2 , April 2022 , pp. 170 - 192
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Michael Coons

The first-named author was supported in part by NSFC grant 11871082 and the second-named author by the FRG grant PS43707 at UNSW.

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