This is certainly not a research level question, but I didn't get an answer to my question 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.