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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   $\chi _n$   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 the Generalized Riemann Hypothesis. This is the first result extending the support outside the  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 [ 22 ,  23 ].

One-Level Density and Non-Vanishing for Cubic L -Functions Over the Eisenstein Field