The first moment of primitive quartic \(L\)-functions with a secondary term

Abstract

Let $q\equiv 3(\mathrm{mod}4)$ be a fixed odd prime power. We study the first moment of central values of primitive quartic Dirichlet $L$-functions over $\F_q[T]$, with fixed genus $g$ and with ${\chi }^2$ primitive. We obtain a main term $C_q{q}^{2g/3}$ and a secondary term $({\alpha }_{r}g+{\beta }_r)q^{g/2}$, with error $O_{q,\eps}(q^{(16/33+\eps)g})$, where $r\equiv g/3+1(\mathrm{mod}4)$. The constants are explicit, with $C_q>0$ and ${\alpha }_r<0$ for every $r$. We use a double Dirichlet series and the functional equation of the quartic Gauss sum generating series to compute the residues and obtain the continuation needed to separate both terms.