One level density of Hecke L-functions associated with cubic characters at prime arguments

We study the one-level density of low-lying zeros of a family of L-functions associated with cubic Hecke characters defined over the Eisenstein field. We show that this family of L-functions satisfies the Katz-Sarnak conjecture for all test functions whose Fourier transforms are supported in (−1, 1), under the Generalized Riemann Hypothesis.