Global functional-equation conjecture for l-adic representations

Let $R$ be an irreducible l-adic representation of $G_{\mathbb{Q}}$ that is de Rham and pure of weight $w\in\mathbb{Z}$. Let $m_i^R$ be the multiplicity of $i$ in its Hodge–Tate multiset, and let $L(\iota R,s)$, $\Lambda(\iota R,s)$, $N(R)$, and $\epsilon(\iota R)$ be the $L$-function, completed $L$-function, conductor, and epsilon factor defined in the source.

Functional-equation conjecture. One has $m_p^R=m_{w-p}^R$ for all $p$, hence $m_{w/2}\equiv\dim V\pmod 2$, and: (1) $L(\iota R,s)$ extends to an entire function except for a single simple pole when $R=\chi_l^{-w/2}$; (2) $\Lambda(\iota R,s)$ is bounded in vertical strips; and (3)

$$\Lambda(\iota R,s)=\epsilon(\iota R)N(R)^{-s}\Lambda(\iota R^\vee,1-s).$$

This combines Fontaine–Mazur with standard analytic conjectures for motivic L-functions. The source gives no general proof or resolution status.