## Mean Value Theorems for $L$-functions over prime polynomials for the rational function field

with J. P. Keating

Acta Arithmetica, Vol. 161, (2013) pp. 371 - 385.

Acta Arithmetica, Vol. 161, (2013) pp. 371 - 385.

**Abstract:**The first and second moments are established for the family of quadratic Dirichlet $L$-functions over the rational function field at the central point $s=\tfrac{1}{2}$ where the character $\chi$ is defined by the Legendre symbol for polynomials over finite fields and runs over all monic irreducible polynomial $P$ of a given odd degree. Asymptotic formulae are derived for fixed finite fields when the degree of $P$ is large. The first moment obtained here is the function field analogue of a result due to Jutila in the number-field setting. The approach is based on classical analytical methods and relies on the use of the analogue of the approximate functional equation for these $L$-functions.