Is there a well known result that states that as $t \to \infty$, 'almost all' zeros of any Dirichlet L function $L(s,\chi)$ lie in the region $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$ for a positive function $\Phi$ "slowly going to zero" as $t\to \infty$, in the sense that the limit of the fraction of the # of zeros of the $L$ function, as $t \to \infty$, within $R\cap \{[0,1] \times[-t,t]\}$, is $1$?
In this case, precise quantifiers above would also be needed. I would be grateful for any possible reference (possibly there is a place in Iwaniec-Kowalski's book where this is mentioned precisely).