Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups

Question

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 $\Gamma_{0}^{+}(N)$.

Answer 1

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, especially section 4.