When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, we know the associated L-function $L(s, \chi)$ can be completed into
$$\Lambda(s,\chi) = \left(\frac{\pi}{q}\right)^{-\frac{1}{2}(s+a)} \Gamma\left(\frac{s+a}{2}\right)L(s, \chi)$$


where $a$ depends of the value of $\chi(-1)$. This completed $L$-function can be holomorphically continued the the whole complex plane (execept for the principal character) and satisfies the functional equation
$$\Lambda(s, \chi) = \frac{\tau(\chi)}{i^a \sqrt{q}}\Lambda(1-s, \overline{\chi})$$


For general number fields $F$, denoting $d$ the degree of the extension over $\mathbb{Q}$, $r\_1$ and $r\_2$ the real and complex embeddings respectively, we can introduce analogously the completed L-function
\begin{equation}
\Lambda(s, \chi) = \left( \frac{2^{r\_1}|d\_F|}{(2\pi)^d} \right)^{s/2} \Gamma\left( \frac{s}{2} \right) \Gamma(s)^{r\_2} L(s, \chi),
\end{equation}


and it satisfies the functional equation, if $\chi$ is a character attached to the ideal $q$:
\begin{equation}
\Lambda(s, \chi) = N(q)^{\frac{1}{2}-s} \Lambda(1-s, \chi).
\end{equation} 


I would like to have a precise reference for L-functions and functional equations in the case of automorphic representations. Let $\pi$ be a self-contragredient cuspidal automorphic representation of $GL(n)$, over a general number field $F$. 



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> Where can I find the precise functional equation satisfies by $L(s, \pi)$, making appear the dependency on the field as well as on the representation $\pi$?
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Functional equation for general number fields
As Kimball mentioned, this is all in Godement-Jacquet's monograph "Zeta Functions of Simple Algebras". For the case $n = 1$, this is just Tate's thesis. When the field is $\mathbb{Q}$, a good reference is Goldfeld-Hundley "Automorphic Representations and $L$-Functions for the General Linear Group".


The story is roughly the following. There is a functional equation of the following form:
\[\Lambda(s,\pi) = \epsilon(s,\pi) \Lambda(1 - s,\widetilde{\pi}).\]
Here $\pi$ is a unitary cuspidal automorphic representation of $\mathrm{GL}\_n(\mathbb{A}\_F)$, where $F$ is a number field, $\widetilde{\pi}$ denotes the contragredient, and $\Lambda(s,\pi) = \prod\_v L\_v(s,\pi\_v)$ is the completed $L$-function of $\pi$ (including the archimedean factors). The epsilon factor $\epsilon(s,\pi)$ factorises as
\[\epsilon(s,\pi) = \epsilon(s,\pi,\psi) = \prod\_v \epsilon\_v(s,\pi\_v,\psi\_v),\]
where $\psi\_v$ is an additive character of $F\_v$ that is the local component of an additive character of $\mathbb{A}\_F$. Each local epsilon factor may depend on $\psi\_v$, but the global epsilon factor is independent of $\psi$.


Let $v$ be a nonarchimedean place of $F$ with associated local field $F\_v$ having ring of integers $\mathcal{O}\_v$, maximal ideal $\mathfrak{p}\_v$, and whose residue field $\mathcal{O}\_v / \mathfrak{p}\_v$ has cardinality $q\_v$. Let $\psi\_v$ be an additive character of $F\_v$. The conductor of $\psi\_v$ is $\mathfrak{p}\_v^{c(\psi\_v)}$, where $c(\psi\_v)$ is the least integer (possibly negative) for which $\psi\_v$ is trivial on $\mathfrak{p}\_v^{c(\psi\_v)}$. The conductor of $\pi\_v$ is $\mathfrak{p}\_v^{c(\pi\_v)}$, where $c(\pi\_v)$ is the least integer (necessarily nonnegative) for which $\pi\_v$ contains a nonzero vector fixed by the congruence subgroup
$$ K\_0(\mathfrak{p}^{c(\pi\_v)}) = \left\{ \begin{pmatrix} a & b \\\ c & d \end{pmatrix} \in \mathrm{GL}\_n(\mathcal{O}\_v) : c \in \mathrm{Mat}\_{1 \times (n - 1)}(\mathfrak{p}\_v^{c(\pi\_v)}), \ d - 1 \in \mathfrak{p}\_v^{c(\pi\_v)} \right\}.$$
Then the local epsilon factor is
\[\epsilon\_v(s,\pi\_v,\psi\_v) = \epsilon\_v\left(\frac{1}{2},\pi\_v,\psi\_v\right) q\_v^{(n c(\psi\_v) - c(\pi\_v))\left(s - \frac{1}{2}\right)}.\]
Suppose that $F\_v$ is an extension of $\mathbb{Q}\_p$. If we choose $\psi\_v$ to be the composition of the standard unramified additive character of $\mathbb{Q}\_p$ with the trace map $\mathrm{Tr}\_{F\_v/\mathbb{Q}\_p}$, then $\epsilon\_v(1/2,\pi\_v,\psi\_v)$ is a complex number of absolute value $1$. For example, if $n = 1$, so that $\pi\_v$ is a character, $\epsilon\_v(1/2,\pi\_v,\psi\_v)$ is a normalised Gauss sum; in general, it can be quite complicated. Moreover, $\mathfrak{p}\_v^{-c(\psi\_v)}$ is the different ideal $\mathfrak{D}\_{F\_v/\mathbb{Q}\_p}$ and $q\_v^{-c(\psi\_v)}$ is the absolute discriminant $\Delta\_{F\_v/\mathbb{Q}\_p} = N\_{F\_v/\mathbb{Q}\_p}(\mathfrak{D}\_{F\_v/\mathbb{Q}\_p})$ of the extension $F\_v/\mathbb{Q}\_p$.


Note that at all but finitely many places, the additive character $\psi\_v$ is unramified ($c(\psi\_v) = 0$) and the representation $\pi\_v$ is unramified ($c(\pi\_v) = 0$).


I haven't yet discussed the archimedean factors. The local $L$-function at an archimedean place $v$ of $F$ will be something of the form
\[L\_v(s,\pi\_v) = \prod\_{j = 1}^{n} \zeta\_v(s + t\_j),\]
for some complex numbers $t\_j$ (with some restrictions on the possible vertical lines that $t\_j$ lies on, since I am assuming that $\pi$ is unitary). Here $\zeta\_v(s) = \pi^{-s/2} \Gamma(s/2)$ for a real place $v$ and $\zeta\_v(s) = 2(2\pi)^{-s} \Gamma(s)$ for a complex place $v$. Finally, the local epsilon factor at an archimedean place will just be something of the form $\epsilon\_v(s,\pi\_v,\psi\_v) = i^k$ for some integer $k$ (in particular, this is independent of $s$). For more details on the archimedean $L$-functions and epsilon factors, see Knapp's paper "Local Langlands Correspondence: the Archimedean Case".


So the global epsilon factor is
\[\prod\_v \epsilon\_v\left(\frac{1}{2},\pi\_v, \psi\_v\right) \prod\_{v \text{ nonarchimedean}} q\_v^{n c(\psi\_v) \left(s - \frac{1}{2}\right)} \prod\_{v \text{ nonarchimedean}} q\_v^{- c(\pi\_v) \left(s - \frac{1}{2}\right)}.\]
The first term is $\epsilon(1/2,\pi)$, the global root number, which is some complex number of absolute value $1$. The second is
\[\Delta\_{F/\mathbb{Q}}^{-n\left(s - \frac{1}{2}\right)},\]
where $\Delta\_{F/\mathbb{Q}}$ is the absolute discriminant of the extension $F/\mathbb{Q}$; this is the norm $N\_{F/\mathbb{Q}}(\mathfrak{D}\_{F/\mathbb{Q}})$ of the different
$$\mathfrak{D}\_{F/\mathbb{Q}} = \prod\_{v \text{ nonarchimedean}} \mathfrak{p}\_v^{-c(\psi\_v)}.$$
The third is
\[N\_{F/\mathbb{Q}}(\mathfrak{q})^{-\left(s - \frac{1}{2}\right)},\]
where
\[\mathfrak{q} = \prod\_{v \text{ nonarchimedean}} \mathfrak{p}\_v^{c(\pi\_v)}\]
is the (relative) conductor of $\pi$ and $N\_{K/\mathbb{Q}}$ denotes the norm.


