$E_a(z,s)$ 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.