Mean value of cubic $L$-funcitons with fixed genus

We investigate the mean value of the first moment of primitive cubic $L$-functions over ${\mathbb{F}}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{χ primitive\ cubic\ genus(χ)=g}}L_q(\frac{1}{2}, χ), \end{equation*} where $L_q(s,\chi )$ denotes the $L$-function associated with primitive cubic character $\chi$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{7}{8}+\epsilon )g}$.