Distribution of zeros of twisted automorphic \(L\)-functions with large level (Q1297938)

Let \(q\) be a prime number and \({\mathcal F}_2(q)\) denote the orthogonal basis of \(S_2(\Gamma_0(q))\) consisting of newforms of weight 2 for the Hecke subgroup \(\Gamma_0(q)\). Let \(m_f(\rho)= m_{f,\chi}(\rho)\) denote the vanishing order of the \(L\)-series attached to \(f\) twisted by a real character \(\chi\) of conductor \(d< \sqrt{q}\), at \(s=\rho\). For any fixed \(s_0\) lying on the critical line \(\operatorname {Re}s=1\) we denote \(R(s_0,\delta)= \sum_{|\rho- s_0|\leq \delta} m_f(\rho)\). We prove that the average (over \(f\in{\mathcal F}_2(q)\)) of \(R(s_0,\delta)\) is proportional to \(\delta\log q\) for sufficiently small \(\delta> 0\) satisfying the condition \(\delta\gg \frac{\log\log q}{\log q}\). Moreover we prove that the average of \(R(s_0,0)\) is \(\ll \log\log q\) where the implied constants in the symbols ``\(\gg\)'' and ``\(\ll\)'' depend on \(s_0\).