Riemann hypothesis for the completed motivic L-function

Fix number fields $K,L$ and a pure motive $M$ of weight $w$ over $K$ with coefficients in $L$. Let $L(s,M)$ be its motivic $L$-function, let $L_{\infty}(s,M)$ be its archimedean factor, and define the completed $L$-function

$$\Lambda(s,M)=N^{s/2}L(s,M)\prod_{\infty}L_{\infty}(s,M).$$

Let $d$ be the dimension of the fixed subspace of the motivic Galois group of $M(-w/2)$, taken to be $0$ if $w$ is odd. The motivic $L$-function conjecture. The function $s^d(1-s)^d\Lambda(s,M)$ extends from $\operatorname{Re}(s)>1$ to an entire function on $\mathbb{C}$ of order $1$ that does not vanish at $s=0,1$; if $M^*$ is the Cartier dual of $M$, then there exists $\epsilon\in\mathbb{C}$ with $|\epsilon|=1$ such that

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

for all $s\in\mathbb{C}$; and all zeros of $\Lambda(s,M)$ lie on the line $\operatorname{Re}(s)=1/2$. These assertions are the expected analytic continuation, functional equation, and Riemann hypothesis for motivic $L$-functions; they generalize the corresponding properties of classical $L$-functions and are presented here as conjectural properties.