TITLE: Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties
QUESTION [9 upvotes]: $\newcommand{\Q}{\Bbb Q}
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\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?

REPLY [4 votes]: $\newcommand{\p}{\mathfrak{p}}$By Theorem 6.3 of this paper by Keith Conrad, strong conjectures about $L(A,s)$ (stronger than GRH for this $L$-function, but still "believable"), imply that
$$
\prod_{N\p\le x}L_{\p}(A,N\p^{-1}) \sim C (\log x)^r
$$
where $r$ is the order of vanishing at $s=1$, which by the usual BSD should be the rank of $A(K)$.
See also this question about the relation between the original and the modern formulation of BSD.