Moments and One level density of sextic Hecke $L$-functions of $\mathbb{Q}(\omega)$

TL;DR

This paper investigates the statistical behavior of sextic Hecke L-functions over () and their zeros, providing new insights into their moments, zero distribution, and non-vanishing properties under GRH.

Contribution

It introduces new results on moments and low-lying zeros of sextic Hecke L-functions over (), including a non-vanishing proportion assuming GRH.

Findings

Derived moments of central values of sextic Hecke L-functions.

Established a one level density result for low-lying zeros.

Proved that at least 2/45 of these L-functions do not vanish at s=1/2 under GRH.

Abstract

In this paper, we study moments of central values of sextic Hecke $L$-functions of $\mathbb{Q}(\omega)$ and one level density result for the low-lying zeros of sextic Hecke $L$-functions of $\mathbb{Q}(\omega)$. As a corollary, we deduce that, assuming GRH, at least $2/45$ of the members of the sextic family do not vanish at $s=1/2$.