Lower order terms for the one-level density of a symplectic family of Hecke L-functions

Research output: Contribution to journalResearch articleContributedpeer-review

Contributors

  • Ezra Waxman - , TUD Dresden University of Technology, Tel Aviv University, Charles University Prague (Author)

Abstract

In this paper we apply the L-function Ratios Conjecture to compute the one-level density for a symplectic family of L-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function f reaches 1, we observe a transition in the main term, as well as in the lower order term. The transition in the lower order term is in line with behavior recently observed by D. Fiorilli, J. Parks, and A. Södergren in their study of a symplectic family of quadratic Dirichlet L-functions [11,12]. We then directly calculate main and lower order terms for test functions f such that supp(fˆ)⊂[−α,α] for some α<1, and observe that this unconditional result is in agreement with the prediction provided by the Ratios Conjecture. As the analytic conductor of these L-functions grow twice as large (on a logarithmic scale) as the cardinality of the family in question, this is the optimal support that can be expected with current methods. Finally, as a corollary, we deduce that, under GRH, at least 75% of these L-functions do not vanish at the central point.

Details

Original languageEnglish
Pages (from-to)447-483
Number of pages37
JournalJournal of Number Theory
Volume221
Publication statusPublished - Apr 2021
Peer-reviewedYes
Externally publishedYes

Keywords

ASJC Scopus subject areas

Keywords

  • Hecke L-functions, One level density, Ratios conjecture