One level density of low-lying zeros of quadratic Hecke $L$-functions of Imaginary Quadratic Number Fields

TL;DR

This paper establishes a one level density result for low-lying zeros of quadratic Hecke L-functions in imaginary quadratic fields with class number one, implying most do not vanish at the critical point.

Contribution

It provides the first one level density result for these L-functions and shows that over 94% of them are non-vanishing at 1/2.

Findings

Over 94% of the L-functions do not vanish at 1/2

First one level density result for quadratic Hecke L-functions in imaginary quadratic fields

Supports non-vanishing conjectures for these L-functions

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 one. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27... \%$ of the $L$-functions under consideration do not vanish at $1/2$.