Zeroes of $L(s,q)=\prod_{\chi(\text{mod}\ q)}L(s,\chi)$ in a rectengle

Question

Denote $$L(s,q)=\prod_{\chi(\text{mod}\ q)}L(s,\chi)$$ Is it true that the number of zeroes of $L(s,q)$ in the rectengle $\{T\leq\Im(s)\leq T+1,\frac{1}{2}<\Re(s)<1\}$ is $\mathcal{O}(\log(qT))$ or even $\mathcal{O}(\log^{\mathcal{O}(1)}(qT))$ ?

Answer 1

If $\chi$ is a primitive character modulo $q$, then we have $$\begin{align} N(T,\chi):&=\#\{\rho: L(\rho,\chi)=0, \ 0\leq \Im(\rho)\leq T, \ 0<\Re(\rho)<1\}\\ &=\frac{T}{2\pi} \log \frac{qT}{2\pi e} +O(\log(qT)).\end{align} $$

From this, we have $$ N(T+1,\chi)-N(T,\chi)=O(\log qT). $$

The number of zeros of $L(s,\chi)$ in your rectangle do not exceed the above. Therefore, it is true that your number is $O(q\log qT)$.