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

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{\chi\ primitive\ quartic\\ \chi^2 primitive\\ genus(\chi)=g}}L_q(\frac{1}{2}, \chi), \end{equation*} where $L_q(s,\chi)$ denotes the $L$-function associated with primitive quartic character $\chi$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{3}{5}+\varepsilon)g}$.