  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 $s=1/2$.
