Asymptotic Zeros of Poincaré Series

We study the zeros of Poincaré series Pk,m for the full modular group. We consider the case where m ~ α k for some constant α> 0. We show that in this case a positive proportion of the zeros lie on the line 1/2 + it. We further show that if α> log (2)/2π then the imaginary axis also contains a positive proportion of zeros. We also give a description for the location of the non-real zeros when α is small.