Moments and one level density of sextic Hecke $L$-functions

Abstract

Let $\omega = \exp (2\pi i/3)$. 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 $s=1/2$.