On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges

We study the pair correlation between zeros of a shifted auxiliary $L$-function attached to a non-CM newform, the scale of which is a fixed constant. We prove an unconditional asymptotic result for the pair correlation and introduce a simplicity hypothesis for the zeros of this function, which if true means that multiple zeros of the original $L$-function cannot be separated by the same fixed distance. Our results provide macroscopic information in contrast to the pair correlation of the original $L$-function which is of microscopic nature.