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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?



Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties
$\newcommand{\p}{\mathfrak{p}}$By Theorem 6.3 of [this paper by Keith Conrad](http://www.math.uconn.edu/~kconrad/articles/eulerprod.pdf), 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](https://mathoverflow.net/questions/239653/clarification-on-the-weak-bsd-conjecture) about the relation between the original and the modern formulation of BSD.


