  In this work, we obtain an asymptotic formula for the twisted mean square of
a Dirichlet $L$-function with a longer mollifier, whose coefficients are also
more general than before. As an application we obtain that, for every Dirichlet
$L$-function, more than 41.72\% of zeros are on the critical line and more than
40.74\% of zeros are simple and on the critical line. These proportions also
improve previous results which were proved only for the Riemann zeta-function.
