On the mean value of generalized Dirichlet L-functions with weight of the character sums

Let p be a prime,. denote a Dirichlet character modulo p. For any integer x (1 <= x <= p - 1), (x) over bar denotes the integer inverse of x such that x (x) over tilde = 1( mod p), we study the following mean value of a kind of character sums with generalized Dirichlet L-functions Sigma chi(-1)=1 |Sigma(p-1)(x=1) chi(x+(x) over bar| |L(1, chi, a)|(2), chi not equal chi 0 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.