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

We obtain an asymptotic formula for the first moment of quadratic Dirichlet $L$–functions over function fields at the central point $s=\frac{1}{2}$. Specifically, we compute the expected value of $L(\frac{1}{2},\chi )$ for an ensemble of hyperelliptic curves of genus $g$ over a fixed finite field as $g\to \mathrm{\infty }$. 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.