  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
polynomials $P$ 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.

Number Theory