The mixed second moment of quadratic Dirichlet L-functions over function fields

Abstract

We investigate the mixed second moment involving the second derivative at the central point of a family of quadratic Dirichlet $L$-functions over ${\mathbb{𝔽}}_q(t)$, associated to the hyperelliptic curves of genus $g$. We compute the full degree five polynomial in the asymptotic expansion of the mixed second moment when the cardinality $q$ of the finite field is fixed and the genus $g$ tends to infinity. This is a partial analogue of classical Ingham’s result about the Riemann zeta function.