The Eisenstein series \begin{equation*} G_{2k}(\tau) = \sum_{(m,n) \in \mathbb{Z} \setminus (0,0)} \frac{1}{(m + n \tau)^{2k}} \end{equation*} is absolutely convergent to a holomorphic function of $\tau$ in the upper half-plane (see https://en.wikipedia.org/wiki/Eisenstein_series).
I am wondering why this function is restricted to the upper half-plane. Is that just a convention, because the information on the lower half-plane is redundant? Or is the analytic continuation not well-defined?
My questions:
- Is there a unique analytic continuation on the whole complex plane (up to poles)?
- If yes, does $G_{2k}(\bar\tau) = \overline{G_{2k}(\tau)}$ hold for the analytic continuation?
- Is each singularity of $G_{2k}(\tau)$ a pole on the real line? (in particular: no essential singularity, no branch point)
- Where are the poles and zeros, of which multiplicity?
I understand that some questions may not have a short answer, but I would like to have an overview of what is well-known to experts in this area.