The real analytic Eisenstein series defined by the Poincare sum
$$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$
for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$ invariant modular function. It has a meromorphic continuation in $s$ but we will not need it for now.
The questions are about the local and global extrema of $E(s,z)$ on the fundamental domain of $SL(2,\mathbb{Z})$. We restrict to the case of $s\in\mathbb{R}$ and $s>1$ for now. Let us start with two observations.
O1: It is clear that the points $z=i$ and $z=e^{\pi i\over 3}$ are local extrema of $E(s,z)$. This is because these points are preserved by finite order subgroups of $SL(2,\mathbb{Z})$ and so the derivatives in $z$ vanish at these points.
O2: Numerically, it appears that $z=i$ and $z=e^{\pi i\over 3}$ are the only local extrema of $E(s,z)$ in the fundamental domain. Furthermore $z=i$ is a saddle point whereas $z=e^{\pi i \over 3}$ is the global minimum.
Here are the questions:
Q1: Is the observation O2 above actually true (known)? In other words, are $z=i$ and $z=e^{\pi i/3}$ the only two extrema for $E(s,z)$ with $s\in \mathbb{R}$ and $s>1$? Furthermore $z=i$ is always a saddle whereas $z=e^{\pi i/3}$ is the global minimum?
Q2: What are some general results on the location of extrema in certain classes of real analytic modular functions? For example for the real part of the Eisenstein series $E(s,z)$ at general complex $s$ and Maass cusp forms, what is known about their extrema? I don't think the simple properties in Q1 will hold for these cases in general.
