The conjectural analytic continuation and functional equation for motivic L-functions

Let $M$ be an object of the category of premotivic structures over $K$ that is $L$-admissible at every prime. Let $L(M,s)$ be its $K\otimes\mathbb{C}$-valued $L$-function, let $\Lambda(M,s)=L_\infty(M,s)L(M,s)$ be its completed $L$-function, and let $M^*=\operatorname{Hom}_K(M,K)$ be the contragredient. The analytic continuation and functional-equation conjecture. The function $L(M,s)$ converges to a holomorphic function on some right half-plane and extends meromorphically to $\mathbb{C}$. Moreover,

$$\Lambda(M,s)=\epsilon(M,s)\Lambda(M^*,1-s),$$

and $\Lambda(M,s)$ is holomorphic if $M$ has no direct summand isomorphic to the unit premotivic structure $K$. This combines the Fontaine–Mazur conjecture with Deligne's conjectured functional equation; the statement concerns the expected analytic properties of motivic $L$-functions and is unresolved in this generality.