  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.
