The Mellin transform of the Whittaker function of an Eisenstein series on GL(2) (with trivial central character) is 
$${L(s+s_1,\chi)L(s+1-s_1,\bar\chi)\over L(2s_1,\chi^2)}$$ 
where s$_1$ and $\chi$ are the data in the Eisenstein series. If we could use s$_1$=1 and $\chi=1$, we'd have your L-function (replacing s with s-1 and getting rid of the denominator). This is exactly where the Eisenstein series has a pole, though. I think you can finesse this issue by using the 0-th coefficient in the Laurent expansion of the Eisenstein series around $s_1=1$ (i.e. ${\rm lim}_{s_1\rightarrow 1} \big(E_{s_1}-{1\over s_1-1}\big)$). I'm pretty sure this works, since the Whittaker function of the residual representation is identically zero, so subtracting it won't change the calculation.

If you had a (nontrivial) Dirichlet L-function, the Eisenstein series doesn't have a pole (well, $\chi^2$ can't be trivial), so it would work fine (if you wanted $L(s-1,\chi)L(s,\bar\chi)$). I am somewhat surprised that the calculation of your L-function is independent of N, though I'm ignorant of such things, I defer to your calculation, but I'd suggest that you double-check.