It is well known that the (real analytic) Eisenstein series is defined, in the slash notation, as follows
$$E\_{s}(\tau) = \sum\limits\_{\gamma\in\Gamma\_{\infty}\backslash\text{SL}(2,\mathbb{Z})}\left.y^{s}\right\vert\_{\gamma},$$
where $s\in\mathbb{C}$ and $\Gamma\_{\infty}$ is the stabilizer of $\text{SL}(2,\mathbb{Z})$. This sum converges when $\text{Re}(s)>1$ and when continued analytically to all of the $s$-plane, it reads
$$E\_{s}(\tau) = y^{s} + \frac{\Lambda(1-s)}{\Lambda(s)}\frac{1}{y^{s-1}} + \sum\limits\_{j=1}^{\infty}\frac{4\sigma\_{2s-1}(j)\sqrt{y}\ K\_{s-\tfrac{1}{2}}(2\pi jy)}{\Lambda(s)j^{s - \tfrac{1}{2}}}\cos(2\pi jx),$$
where $\sigma\_{2s-1}(j)$ is the divisor sigma function, $K\_{\nu}(z)$ is the modified Bessel function of the second kind, and $\Lambda(s)$ is the completed zeta function as a function of $2s$, $\Lambda(s) = \pi^{-s}\Gamma(s)\zeta(2s)$ ,that obeys the functional equation $\Lambda\left(\tfrac{1}{2}-s\right) = \Lambda(s)$. This real analytic Eisenstein series is an eigenfunction of the Laplacian defined on the space $\text{SL}(2,\mathbb{Z})\backslash\mathbb{H}$, where $\mathbb{H}$ is the upper half-plane. I have the following questions:


1. Does there exist explicit formulae in literature for the analytic continuation of $E\_{s}(\tau)$ defined on the space with Hecke and Fricke subgroup quotients, i.e. $\Gamma\_{0}(N)\backslash\mathbb{H}$ and $\Gamma\_{0}^{+}(N)\backslash\mathbb{H}$?
2. I suspect the analytic continuation formulae to contain certain L-functions $L^{\*}(s)$ akin to how $\Lambda(s)$ shows up in the above definition. If they don't then is it possible to reexpress these formulae in terms of some nice L-functions?


Here, the Hecke group $\Gamma\_{0}(N)$ is a discrete subgroup of $\text{SL}(2,\mathbb{Z})$ and is a subgroup of index $2$ in the Fricke group $\Gamma\_{0}^{+}(N)$.



Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups
One natural generalization of $E\_s$ to $\Gamma\_0(N) \backslash \mathbb{H}$ is the family of Eisenstein series, $$E\_{s; \chi\_1, \chi\_2}(\tau) = \sum\_{\gamma \in \Gamma\_{\infty} \backslash \mathrm{SL}\_2(\mathbb{Z})} \chi\_1(c) \chi\_2(d) (q\_2 y)^s \Big| \gamma, \quad \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$$ where $\chi\_1$ and $\chi\_2$ are Dirichlet characters modulo $q\_1$, $q\_2$ satisfying $\chi\_1(-1) = \chi\_2(-1)$. These are automorphic on $\Gamma\_0(q\_1 q\_2)$ with Nebentypus character $\chi\_1 \overline{\chi\_2}$.


The series $E\_{s; \chi\_1, \chi\_2}$ have an analytic continuation and have natural completions $E^\*$, with the completed $L$-function of $\chi\_1 \chi\_2$ playing the role of $\Lambda$. The functional equation should now relate Eisenstein series attached to different characters, taking the form $$E\_{s; \chi\_1, \chi\_2}^\* = E\_{1-s, \overline{\chi\_2}, \overline{\chi\_1}}^\*.$$ For a detailed reference (also including weight other than zero) see the paper [*Explicit calculations with Eisenstein series* by M. Young](https://arxiv.org/abs/1710.03624), especially section 4.


