The first moment of central value of primitive quartic $L$-functions with fixed genus

Abstract

We investigate the mean value of the first moment of primitive quartic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{χ primitive\ quartic\\ χ^2 primitive\\ genus(χ)=g}}L_q(\frac{1}{2}, χ), \end{equation*} where $L_q(s,χ)$ denotes the $L$-function associated with primitive quartic character $χ$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{3}{5}+\varepsilon)g}$.