Zeros of Dirichlet $L$-functions near the critical line

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 $38.2\mathrm{\%}$ of zeros of the primitive Dirichlet $L$-functions of modulus $q$ lie on the critical line.