  We prove an upper bound on the density of zeros very close to the critical
line of the family of Dirichlet $L$-functions of modulus $q$ at height $T$. To
do this, we derive an asymptotic for the twisted second moment of Dirichlet
$L$-functions uniformly in $q$ and $t$. As a second application of the
asymptotic formula we prove that, for every integer $q$, at least $38.2\%$ of
zeros of the primitive Dirichlet $L$-functions of modulus $q$ lie on the
critical line.
