If $A$ is an abelian variety over a global field $K$, its $L$-function
$L(A,s)$ is the object of the famous Birch and Swinnerton-Dyer conjecture.
It is known that the case where $K$ has positive characteristic is easier
than the one where $K$ is a number field: for example, it is known in the
positive characteristic case that the rank of $A$ is bounded above by the
analytic rank. I will revisit this result by giving a formula for $L(A,s)$
in terms of zeta functions of motives associated to $A$, defined over the
field of constants of $K$ (a finite field). Further applications will be
given if time permits.