Pair correlation of zeros of \(\Gamma_1(q)\) L-functions (Q2163302)

In the paper under review, the authors examine the pair correlation function of a family of \(L\)-functions attached to automorphic forms on \(\mathrm{GL}(2)\). By assuming the Riemann hypothesis, they evaluate the pair correlation function in the range \(|\alpha |< 2\), where the off-diagonal terms also contribute to the main term. As a consequence, it is shown that the proportion of simple zeros of all \(\Gamma_1(q)\) \(L\)-functions is greater than or equal to \(0.9350.\)