  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. In this paper, we study the mean value properties of
the generalized Dirichlet $L$-functions, and obtain several sharp asymptotic
formulae by using analytic method.
