Functional equation of the $L$-function of a motive

Question

Suppose $M$ is a pure motive over $\mathbb{Q}$, and let $L(M,s)$ be its $L$-function obtained as product of local $L$-factors at non-archimedean places, while let $L_\infty(M,s)$ be the product of $L$-factors at archimedean places. The total $L$-function is \begin{equation} \Lambda(M,s)=L(M,s)L_\infty(M,s) \end{equation} The total $L$-function is conjectured to satisfy a functional equation \begin{equation} \Lambda(M,s)=\epsilon(M,s)\Lambda(M^\vee,1-s) \end{equation} where $M^\vee$ is the dual of $M$. Could anyone discuss some intuitions (examples) why this should be true, e.g., why we need to take the dual of $M$? Why it is $1-s$ on the right hand side instead of say $n-s, n \in \mathbb{Z}_+$ in general?

Answer 1

Some clarification is needed in the definition of your motif $M$ and its $L$-function. For two number fields, $F$ and $E$, recall that the (still conjectural) "mixed motives" defined over $F$, with coefficients in $E$, are supposed to be the objects $M$ of a certain category $\mathcal M\mathcal M_F (E)$ with a "unit object" $1_{F,E}$. The dual motif $M^*$ of $M$ should be $Hom (M,1_{F,E})$, and for all $i\in \mathbf N$, the "motivic cohomology groups" should be defined as $H_{\mathcal M\mathcal M}^i (F,M):= Ext_{\mathcal M\mathcal M}^i (1_{F,E},M)$. In practice, one works only with a system of étale, Betti and de Rham "realizations" (the sanksrit word "avatars" would be more suggestive) of the hypothetical motif $M$, together with comparison isomorphisms. For $E=\mathbf Q$, an important particular example is the Tate motif $\mathbf Q(m), m\in \mathbf Z$, which is a pure motif over $F$ of weight $-2m$, whose étale avatars $M_{\mathcal L}$, for all places $\mathcal L$ of $E$ above prime numbers $\mathcal l$, are the Tate twists $\mathbf Q_\mathcal l (m)$; whose Betti avatars, for all the archimedean places of $F$, are $(2\pi i)^m \mathbf Q$; whose de Rham avatar is $F$ equipped with a certain filtration. Here note that for $m\in \mathbf N,\mathbf Q(m)= \mathbf Q(1)^{\otimes m}$, and the dual of $\mathbf Q(m)$ is $\mathbf Q(-m)$. The $L$-function $L(\mathbf Q(m),s)$ is the shifted Dedekind zeta function $\zeta_F (s+m)$. To avoid sinking an already overloaded boat, I refer to [Pu] below for more details.

With the example of the Tate motives in mind, Deligne (1979) has given an axiomatic definition of the category of "Deligne 1-motives" and of the attached $L$-functions, and has conjectured the functional equation they should satisfy (as written in your post). When $E=\mathbf Q$, no wonder then that the dual motif enters the game, as well as the change of variables $s \to 1-s$. Besides,if you look at Tate's approach to Hecke- zeta functions using Fourier analysis in number fields (his 1950 thesis), the functional equation is written $\zeta(f,c)=\zeta(\hat f, \hat c)$, where $\hat f\in L^1(\hat G)$ is the Fourier transform of $f\in L^1( G)$ (see his main theorem 4.4.1 for the precise statements). When going back to the Dedekind functions $\zeta(s,\chi)$, the passage between the two "dual" expressions above corresponds to $(s, \chi) \to(1-s, \chi^{-1})$ .

[Pu] "The Bloch-Kato conjecture for the Riemann zeta function", Proceedings of the Pune 2012 Conference, London Math. Soc. LNS 418 (2015). See in particular chapters 1 and 9.