# Functional equation for general number fields

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 $$$$\Lambda(s, \chi) = \left( \frac{2^{r_1}|d_F|}{(2\pi)^d} \right)^{s/2} \Gamma\left( \frac{s}{2} \right) \Gamma(s)^{r_2} L(s, \chi),$$$$

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

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 $$\pi$$?

• This amounts to knowing archimedean L-factor as well as the global epsilon factor. Have you looked in Godement-Jacquet? Jan 14, 2019 at 21:31

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.

• 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) Jan 16, 2019 at 1:13
• 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! Jan 16, 2019 at 3:59