One level density of low-lying zeros of quadratic and quartic Hecke $L$-functions

TL;DR

This paper investigates the distribution of low-lying zeros of quadratic and quartic Hecke L-functions over the Gaussian field, providing density results and implications for non-vanishing at the central point.

Contribution

It establishes new one level density results for these families, leading to improved non-vanishing proportions at the central point.

Findings

At least 94.27% of quadratic family members do not vanish at the center.

At least 5% of quartic family members do not vanish at the center.

Provides density results for low-lying zeros of Hecke L-functions.

Abstract

In this paper, we prove some one level density results for the low-lying zeros of famliies of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that, respectively, at least $94.27 \%$ and $5\%$ of the members of the quadratic family and the quartic family do not vanish at the central point.