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% of zeros of the primitive Dirichlet L-functions of modulus q lie on the critical line.