Non-vanishing of primitive cubic, quartic and sextic L-functions

Abstract

We establish lower bounds for the non-vanishing proportions of primitive cubic, quartic and sextic Dirichlet $L$-functions over $\F_q[T]$ in the non-Kummer setting. Adapting truncated-exponential techniques for shifted moments, we improve the cubic lower bound of David, Florea and Lalín and obtain positive lower bounds for quartic and sextic families. For each family, the lower limit of the non-vanishing proportion exceeds $e^{-e^{53}}$ as the conductor degree tends to infinity with $q$ fixed. We use a common choice of truncation parameters and explicitly control the exceptional sets. The cubic argument uses the published twisted first moment. We also establish uniform twisted first moments in the quartic and sextic cases, yielding an asymptotic formula for the sextic first moment.