Pair correlation of zeros of Γ1(q) L-functions

We study the analogue of the Montgomery’s pair correlation function F ( α ) for a family of Γ 1 ( q ) L -functions. Assuming the Generalized Riemann Hypothesis, we evaluate this analogous pair correlation function in the range | α | < 2 , where the off-diagonal terms also contribute to the main term. As applications, we obtain that more than 93.5% of the low-lying zeros of Γ 1 ( q ) L -functions are simple, and we achieve upper and lower bounds for the number of pairs of zeros which are closer than β times the average spacing apart.