On the mean value of the generalized Dirichlet L-functions with the weight of the Gauss Sums

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$ with $\sigma>1$ and $t$ both real. It can be extended to all $s$ by analytic continuation. For any integer $m$, the famous Gauss sum $G(m,\chi)$ is defined as follows: $$G(m,\chi)=\sum_{a=1}^{q}\chi(a)e\left(\frac{am}{q}\right), $$ where $e(y)=e^{2\pi iy}$. The main purpose of this paper is to use the analytic method to study the mean value properties of the generalized Dirichlet $L$-functions with the weight of the Gauss Sums, and obtain a sharp asymptotic formula.