of $s\in\mathbb{C}$ by a functional equation in $s\mapsto \frac{n+1}{2}-s$. This series belongs to $C_{\kappa}^{\infty}(\Gamma, {\operatorname{\varphi}}^{-1})$ and is normalised by a product of Dirichlet $L$-functions as follows. Let ${\mathfrak}{a}$ be any integral ideal and define $$\begin{aligned}
    \begin{split}
        L_{{\mathfrak}{a}}(s, {\operatorname{\varphi}}) :&= \prod_{p\nmid{\mathfrak}{a}}(1-{\operatorname{\varphi}}^*(p)p^{-s})^{-1}; \\
        \Lambda_{{\mathfrak}{a}}^{n, \kappa}(s, {\operatorname{\varphi}}) :&= \begin{cases}
            L_{{\mathfrak}{a}}(2s, {\operatorname{\varphi}})\displaystyle\prod_{i=1}^{[n/2]}L_{{\mathfrak}{a}}(4s-2i, {\operatorname{\varphi}}^2) &\text{if $\kappa\in\mathbb{Z}$}, \\
            \displaystyle\prod_{i=1}^{[(n-1)/2]}L_{{\mathfrak}{a}}(4s-2i+1, {\operatorname{\varphi}}^2) &\text{if $\kappa\notin\mathbb{Z}$}.
            \end{cases}
    \end{split}\end{aligned}$$ The normalised Eisenstein series is given by $$\begin{aligned}
    \mathcal{E}(z, s; \kappa, {\operatorname{\varphi}}, \Gamma') := \overline{\Lambda_{{\mathfrak}{y}}^{n, \kappa}(s, \bar{{\operatorname{\varphi}}})}E(z, \bar{s}; \kappa, {\operatorname{\varphi}}, \Gamma'). \label{normalisedeisen}\end{aligned}$$

Set $\eta := \psi\bar{\chi}\rho_{\tau}$. In this setting, the integral expression of [@Shimuraexp (4.1)] becomes: $$\begin{aligned}
    \begin{split}
        L_{\psi}(s, f, \bar{\chi}) &= \left[\Gamma_n\left(\tfrac{s-n-1+k+\mu}{2}\right)2c_f(\tau, 1)\right]^{-1}N({\mathfrak}{b})^{\frac{n(n+1)}{2}}|4\pi\tau|^{\frac{s-n-1+k+\mu}{2}} \\
    &\times \left(\tfrac{\Lambda_{{\mathfrak}{c}}}{\Lambda_{{\mathfrak}{y}}}\right)\left(\tfrac{2s-n}{4}\right)\prod_{q\in\mathbf{b}}g_q\left((\psi^{{\mathfrak}{c}}\bar{\chi}^*)(q)q^{-s}\right)\langle f, \theta_{\bar{\chi}}\mathcal{E}(\cdot, \tfrac{2s-n}{4})\rangle_{{\mathfrak}{y}} V,
    \end{split}\label{RS}\end{aligned}$$ in which $\Lambda_{{\mathfrak}{a}}(s) := \Lambda_{{\mathfrak}{a}}^{n, k-\frac{n}{2}-\mu}(s, \eta)$; ${\mathfrak}{y}:= {\mathfrak}{c}\cap(2{\mathfrak}{t}{\mathfrak}{f}^2)$; $\mathbf{b}$ is the finite set of primes $q\nmid{\mathfrak}{c}$ such that ${\operatorname{ord}}_q(|\tau|)\neq 0$ and $g_q\in\mathbb{Q}[t]$ satisfy $g_q(0) = 1$; $$\mathcal{E}(z, s) := \mathcal{E}\left(z, s; k-\tfrac{n}{2}-\mu, \bar{\eta}, \Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{by}]\right);$$ and $V := {\operatorname{Vol}}(\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{by}]{\backslash}\mathbb{H}_n)$.

$p$-stabilisation {#pstabsection}
=================

If $p$ divides the level of $f$ then the behaviour of the $p$th Hecke operator becomes simpler; this is a key tool in the construction of the $p$-adic $L$-function. For full generality we begin with a form $f$, assume that $p\nmid{\mathfrak}{c}$ does not divide the level, 