Using estimates of character sums and the analytic method, the authors study the first power mean of Dirichlet \(L\)-functions with the weight of trigonometric sums and obtain a sharper asymptotic formula for the hybrid mean value \[ \sum_{\chi\not =\chi_0}\left|\sum_{a=1}^{p-1}(\frac{f(a)}{p})\right|^2\,L(1, \chi). \]