ZEROS OF THE EISENSTEIN SERIES E2

In this paper we investigate the zeros of the Eisenstein series E-2. In particular, we prove that E-2 has infinitely many SL2(Z)-inequivalent zeros in the upper half-plane Si, yet none in the standard fundamental F. Furthermore, we go on to investigate other fundamental regions in the upper half-plane n for which there do or do not exist zeros of E-2. We establish infinitely many such regions containing a zero of E-2 and infinitely many which do not.