Moments of Dirichlet L-functions with prime conductors over function fields

We compute the second moment in the family of quadratic Dirichlet L–functions with prime conductors over Fq[x] when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an asymptotic formula with the leading order term for the mean value of the derivatives of L–functions associated to quadratic twists of a fixed elliptic curve over Fq(t) by monic irreducible polynomials. As a corollary, we prove that there are infinitely many monic irreducible polynomials such that the analytic rank of the corresponding twisted elliptic curves is equal to 1.