## The Mean Value of $L(\tfrac{1}{2},\chi)$ in the Hyperelliptic Ensemble

with J. P. Keating

Journal of Number Theory, Volume 132, Issue 12 (2012) pp. 2793-2816.

Journal of Number Theory, Volume 132, Issue 12 (2012) pp. 2793-2816.

**Abstract:**We obtain an asymptotic formula for the first moment of quadratic Dirichlet $L$-functions over the rational function field at the central point at $s=\tfrac{1}{2}$. Specifically, we compute the expected value of $L(\tfrac{1}{2},\chi)$ for an ensemble of hyperelliptic curves of genus $g$ over a fixed finite field as $g\rightarrow\infty$. Our approach relies on the use of the analogue of the approximate functional equation for such $L$-functions. The results presented here are the function field analogues of those obtained previously by Jutila in the number-field setting and are consistent with recent general conjectures for the moments of $L$-functions motivated by Random Matrix Theory.