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

In this paper, we evaluate explicitly certain quadratic Hecke Gauss sums of $\mathbb{Q}(\omega ),\omega =\exp \left(\frac{2\pi i}{3}\right)$. As applications, we study the moments of central values of quadratic Hecke $L$-functions of $\mathbb{Q}(\omega )$, and establish quantitative non-vanishing result for the $L$-values. We also establish an one level density result for the low-lying zeros of quadratic Hecke $L$-functions of $\mathbb{Q}(\omega )$.