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

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*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 $\Gamma_{0}^{+}(N)\backslash\mathbb{H}$?

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