$\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\F}{\Bbb F} \newcommand{\p}{\mathfrak{p}} $ Let $A$ be an abelian variety over a number field $F$. It is expected that the $L$-function of $A$ has analytic continuation to $\Bbb C$ and satisfies a functional equation relating $s$ to $2-s$. In that setting, the (generalized) Birch–Swinnerton-Dyer conjecture states that $$\mathrm{ord}_{s=1}(L(A_{/F},s)) = \mathrm{rk}_{\Z}(A(F)) =: r.$$

Originally, the conjecture for an *elliptic curve* $E$ over $\Q$ was
$$\exists C>0,\quad
\prod_{p \leq x} \dfrac{|E(\F_p)|}{p} \sim C \;\mathrm{log}(x)^r
\qquad (x \to \infty).$$

My question is to know what is the analogue of the original conjecture, in the framework of abelian varieties over number fields.

My first guess would to replace to LHS by $$\prod_{N(\p) \leq x} L_{\p}(A_{/F}, N(\p)^{-1}),$$ where $L_{\p}(A_{/F},s)$ is the local factor of the L-function of $A$ at $\p$. But I'm not sure what the RHS should be. Typically, how does it depend on the dimension of $A$ or on the degree of the number field?