On the mean value of generalized Dirichlet $${\varvec{L}}$$-functions with weight of the character sums

Let p be a prime, $$\chi $$ denote a Dirichlet character modulo p. For any integer x ( $$1\le x\le p-1$$ ), $${\bar{x}}$$ denotes the integer inverse of x such that $$x{\bar{x}}\equiv 1(\bmod \, p)$$ , we study the following mean value of a kind of character sums with generalized Dirichlet L-functions $$\begin{aligned} {\mathop {\mathop {\sum }\limits _{\chi (-1)=1}}\limits _{\chi \ne \chi _0}} \left| \sum _{x=1}^{p-1}\chi (x+{\bar{x}})\right| ^2|L(1,\chi ,a)|^2, \end{aligned}$$ where $$\chi _0$$ is the principal character modulo p, and $$L(1,\chi ,a)$$ is the generalized Dirichlet L-functions. In this paper, we will use the analytic method and get a sharp asymptotic formula.