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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 $F$.

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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    $\begingroup$ This amounts to knowing archimedean L-factor as well as the global epsilon factor. Have you looked in Godement-Jacquet? $\endgroup$ – Kimball Jan 14 at 21:31
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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.

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    $\begingroup$ Also I would mention Neukrich, algebraic number theory does the full proof of the functional equation of Dedekind and Hecke and Artin L-functions (the latter requires class field theory, so 100 more pages) using the traditional approach of theta series (it takes more than 100 pages) $\endgroup$ – reuns Jan 16 at 1:13
  • $\begingroup$ Thanks! Great answer with all the details, thanks a lot for providing such a synthesis of perhaps standard facts yet not easily accessible in a single place of the litterature! $\endgroup$ – TheStudent Jan 16 at 3:59

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