f, g\rangle := {\operatorname{Vol}}(\Gamma{\backslash}\mathbb{H}_n)^{-1}\int_{\Gamma{\backslash}\mathbb{H}_n}f(z)\overline{g(z)}\Delta(z)^{\kappa}d^{\times}z$$ in which $$\Delta(z) := \det({\operatorname{\mathfrak{Im}}}(z)), \hspace{15pt} d^{\times}z := \Delta(z)^{-n-1}\bigwedge_{i\leq j}(dx_{ij}\wedge dy_{ij}),$$ for any $z = (x_{ij}+iy_{ij})_{i, j=1}^n\in\mathbb{H}_n$. This integral is convergent whenever one of $f, g$ belongs to $\mathcal{S}_{\kappa}(\Gamma, \psi)$.

Complex $L$-function {#Lfunsection}
====================

The standard complex $L$-function is defined in this section and it is ended by stating the known Rankin-Selberg integral expression.

For any Hecke character ${\operatorname{\varphi}}$ of $\mathbb{Q}$ let ${\operatorname{\varphi}}^*(p) = {\operatorname{\varphi}}^*(p\mathbb{Z})$ denote the associated ideal character. Though the integral expression can be stated for any half-integral weight $k$, we take $k\geq n+1$ to ease up on notation – we shall be making this assumption later on anyway. For a prime $p$, the association of the Satake $p$-parameters – $(\lambda_{p, 1}, \dots, \lambda_{p, n})\in\mathbb{C}^n$ – to a non-zero Hecke eigenform $f\in\mathcal{S}_k(\Gamma, \psi)$ is well-known (see, for example, [@Shimurahalf p. 46]). The standard $L$-function of $f$, twisted by a Hecke character $\chi$ of conductor ${\mathfrak}{f}$, is then defined by $$\begin{aligned}
    L_p(t) :&= \begin{cases} \displaystyle\prod_{i=1}^n (1-p^n\lambda_{p, i}t) &\text{if $p\mid{\mathfrak}{c}$}, \\ 
        \displaystyle\prod_{i=1}^n (1-p^n\lambda_{p, i}t)(1-p^n\lambda_{p, i}^{-1}t) &\text{if $p\nmid{\mathfrak}{c}$}; \end{cases} \\
    L_{\psi}(s, f, \chi) :&= \prod_p L_p\left((\psi^{{\mathfrak}{c}}\chi^*)(p)p^{-s}\right)^{-1},\end{aligned}$$ in which $$\psi^{{\mathfrak}{c}}(x) := \left(\frac{\psi}{\psi_{{\mathfrak}{c}}}\right)(x).$$ The Rankin-Selberg integral expression, (4.1) in [@Shimuraexp p. 342], is given there in generality; we restate 