Zeros of the Eisenstein series $E_2$

Abstract:

In this paper we investigate the zeros of the Eisenstein series $E_2$. In particular, we prove that $E_2$ has infinitely many $\operatorname {SL}_2(\mathbb {Z})$-inequivalent zeros in the upper half-plane $\mathfrak {H}$, yet none in the standard fundamental $\mathfrak {F}$. Furthermore, we go on to investigate other fundamental regions in the upper half-plane $\mathfrak {H}$ 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.