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$E_a(z,s)$ denotes the Eisenstein series expanded at the cusp $a$. For each cusp $a=\frac{u}{w}$ of $\Gamma_0(N)$, we define the Eisenstein series $ \begin{eqnarray}E_a(z,s)&=&\sum_{\tau\in\Gamma_a\backslash\Gamma}\Im(\sigma^{-1}_a\tau z)\\&=&\delta_{a\infty}y^s+\sqrt{\pi}\frac{\Gamma(s-\dfrac{1}{2})}{\Gamma(s)}\rho_a(s,0)y^{1-s}\\&+&\frac{2\pi^s\sqrt{y}}{\Gamma(s)}\sum_{m\neq0}|m|^{s-\frac{1}{2}}\rho_a(s,m)K_{s-\dfrac{1}{2}}(2\pi|m|y)e(mx), \end{eqnarray} $ where $\rho_a(s,m)=\left(\frac{(w,\frac{N}{w})}{wN}\right)^s\sum_{(\gamma,\frac{N}{w})=1}\frac{1}{\gamma^{2s}}\sum_{\delta(\bmod \gamma w),(\delta,\gamma w)=1, \delta \gamma\equiv u\left(\bmod(w,\dfrac{N}{w})\right)}e(-m\dfrac{\delta}{\gamma w}).$

I would like to know what is the answer or relation between Riemann's $\zeta(s)$ of the following two L-series $\sum_{n>0}\dfrac{\rho_a(s,n)}{n^w}$ and $\sum_{n>0}\dfrac{\widetilde{\rho_a(s,n)}}{n^w}$ where $\widetilde{\rho_a(s,n)}=\dfrac{\rho_a(s,n)\zeta(2s)}{\sigma_{1-2s}(m)}$. Thanks.

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    @nglineline, Basically, for the $N=1$ case, you'll end up with something like ${\zeta(w+s)\zeta(w+1-s)\over\zeta(2s)}$ For general $N$, $\zeta$ will be replaced by a Dirichlet $L$-function, and there will probably be some additional nuisance factors. Off the top of my head, I can't derive the correspondence between the Dirichlet character and the cusp (though I'm sure cusps in $\Gamma_0(N)$ correspond to Dirichlet characters of order $N$).2012-06-27

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I only answer this because you haven't received an answer. You are asking about the scattering matrix of Eisenstein series (ramified situation).

These computations have been performed by Huxley in his article Scattering matrices for congruence subgroups.

I have obtained similar results in my thesis for global fields, but on $GL(2)$. They translate not that easily. It is in an adelic setting and I restrict attention to rather specific Eisenstein series. The usual Eisenstein series on GL(2) can be obtained from these.

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    Garrett does it [here](http://www-users.math.umn.edu/~garrett/m/mfms/notes_2013-14/12_2_transition_Eis.pdf) : finding the Hecke eigenvalues thus the Euler product for some of those real analytic Eisenstein series2017-11-29