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One-level density and non-vanishing for cubic $L$-functions over the Eisenstein field

We study the one-level density for families of L-functions associated with cubic Dirichlet characters defined over the Eisenstein field. We show that the family of $L$-functions associated with the cubic residue symbols $χ_n$ with $n$ square-free and congruent to 1 modulo 9 satisfies the Katz-Sarnak conjecture for all test functions whose Fourier transforms are supported in $(-13/11, 13/11)$, under GRH. This is the first result extending the support outside the \emph{trivial range} $(-1, 1)$ for a family of cubic L-functions. This implies that a positive density of the L-functions associated with these characters do not vanish at the central point $s=1/2$. A key ingredient in our proof is a bound on an average of generalized cubic Gauss sums at prime arguments, whose proof is based on the work of Heath-Brown and Patterson.