Moments of Artin–Schreier <i>L</i>-functions

Abstract We compute moments of L-functions associated to the polynomial family of Artin–Schreier covers over $\mathbb{F}_q$, where q is a power of a prime p &amp;gt; 2, when the size of the finite field is fixed and the genus of the family goes to infinity. More specifically, we compute the $k{\text{th}}$ moment for a large range of values of k, depending on the sizes of p and q. We also compute the second moment in absolute value of the polynomial family, obtaining an exact formula with a lower order term, and confirming the unitary symmetry type of the family.