I want to do some computations which require the functional equations of the following real analytic Eisenstein series of weight $k$: $$E_k(z, s)=\sum_{(c, d) \in \mathbf{Z}^2 \backslash(0,0)} \frac{y^s}{(c z+d)^k|c z+d|^{2 s}},$$ but I couldn't find any related reference.
In standard references, for example:
- Bump, Automorphic Forms and Representations, it only deals with $k=0$ example;
- Miyake, Modular Forms, in the last chapter, last section, it computes in great detail the Fourier expansion of this Eisenstein series (following Shimura), but without discussion on the functional equation.
Could anybody tell me some other related literature? Thanks a lot in advance!
(Posting this as an answer because I just bountied a post and don't have enough to comment) Try checking out these to begin with:
https://arxiv.org/pdf/2306.10696.pdf and https://arxiv.org/pdf/2310.06284.pdf
https://www-users.cse.umn.edu/~garrett/m/mfms/notes_c/simplest_eis.pdf
https://people.mpim-bonn.mpg.de/zagier/files/scanned/IntroductionToModularForms/fulltext.pdf
Was that what you were looking for?
How about?
https://link.springer.com/article/10.1007/s40687-018-0151-3
In the above one they deal with a generalization of the following Eisentsein series you have $$E_{k-1, 2s-1}(z)=\sum_{(c, d) \in \mathbf{Z}^2 \backslash(0,0)} \frac{\ln\mid q\mid}{(c z+d)^{k}(c \bar{z}+d)^{2s}},$$
multiplied by some constant term dependant on $s$ and $k$, with $q=e^{2\pi i \tau}$