In Peter Schneider's paper "Introduction to the Beilinson conjectures", he lists some hypothesis (conjectures) on the $L$-function associated to the pure motive $H^i(X)$ where $X$ is a smooth projective variety defined over $\mathbb{Q}$.
$L(H^i(X),s)$ converges absolutely for $\text{Re}\,s >1+i/2$, hence it does not vanish in this region.
$L(H^i(X),s)$ has a meromorphic continuation to the whole complex plane, and the only possible pole occurs at $s=1+i/2$ for even $i$.
$L(H^i(X),1+i/2) \neq 0$, (I guess he means when $s=1+i/2$ is not a pole).
$\Lambda(H^i(X),s):=L(H^i(X),s)\cdot L_{\infty}(H^i(X),s)\cdot(\text{exponential factor})$ has a functional equation with respect to the operation $s \mapsto i+1-s$, where $L_\infty(H^i(X),s)$ is the Euler factor associated to the archimedean place of $\mathbb{Q}$.
For a variety $Y$ defined over a number field $K$, its Euler factor at a non-archimedean place is defined similarly.
Question 1. How to define the Euler factor at a archimedean place?
For a real place of $K$, there is still an involution $F_{\infty}$ induced by complex conjugation on the $\mathbb{C}$-valued points $Y(\mathbb{C})$, and I guess the Euler factor is defined as the $\mathbb{Q}$-case? But how to define the Euler factor for a (conjugated pair of) complex place(s)?
Question 2. I guess the four hypothesis are also expected for the $L$-function of $H^i(Y)$. At current stage, for which (class of) smooth projective varieties (over a number field) has these hypothesis been proved?
