Zeros of the Eisenstein series $E_2$

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.