This is certainly not a research level question, but I didn't get an answer to my [question](https://math.stackexchange.com/questions/1488319/how-are-holomorphic-and-real-analytic-eisenstein-series-related) on MSE, so here goes:


The holomorphic Eisenstein series can be given as
$$G\_{2k}(z)=\sum\_{(c,d)\in{\bf Z}^2\backslash(0,0)}\frac{1}{(cz+d)^{2k}}$$
while the real-analytic Eisenstein series is
$$E(z,s)=\frac{1}{2}\sum\_{(c,d)=1}\frac{y^s}{|cz+d|^{2s}}$$
satisfying the relation
$$2\zeta(2s)E(s,z)=\sum\_{(c,d)\in{\bf Z}^2\backslash(0,0)}\frac{\text{Im}(z)^s}{|cz+d|^{2s}}.$$
They look to be very similar, but have different analytic behaviours. I haven't found a discussions on how they are related, so I'm hoping to find some answers.



How are holomorphic and real-analytic Eisenstein series related?
[Comment reposted as an answer]


Up to the scaling by $2 \zeta(2s)$, which is just a matter of conventions, both are special cases of a single more general object: the series
$$ E\_k(z, s) = \sum\_{(c, d) \in \mathbf{Z}^2 \setminus (0,0)} \frac{y^s}{(cz + d)^k |cz + d|^{2s}}, $$
which converges for $Re(s) \gg 0$ (actually for $k + 2 Re(s) > 2$ if I remember correctly) and has meromorphic continuation to all $s \in \mathbf{C}$ (analytic if $k \ne 0$). 


For a fixed value of $s$, the function $E\_k(-, s)$ transforms like a modular form of weight $k$ under the modular group (although it is not usually holomorphic in the $z$ variable). 


