In 'Remarks on Weil's quadratic functional..' p.191 Bombieri claims any given $L$-function $L(s,\chi)$ has at least
$$\big(\frac{1}{\pi}+o(1)\big)R\log R$$
zeroes in a disk $|s|<R$. Is there a reference and/or a proof of this statement that I can cite?



Asymptotic for zeroes of $L(s,\chi)$ in a disk $|s|<R$
I doubt that this is what he's claiming exactly, since if you position your disk in the strip $\Re s > 1$ then the count is zero. I realize now that most likely the disk is expanding (that is $R$ goes to infinity) in that case this follows just from the usual count of zeros which is known to be $(1/2\pi) T \log T + (\text{easy small stuff}) + O(\log T)$ as $T \rightarrow \infty$. The conductor of the L-function also has to be fixed. 


