Hasse-Weil conjecture (reviewed)

The Hasse-Weil conjecture predicts that the L-function $L(A,s)$ of a (positive-dimensional) abelian variety $A$ over a number field $K$ has an analytic continuation to $\C$ with no poles in the critical strip and that it satisfies its functional equation; equivalently, $L(A,s)$ lies in the Selberg class.

The Hasse-Weil conjecture is implied by modularity and is thus known to hold, for example, when $K=\Q$ and $A$ is an elliptic curve or an abelian variety of $\GL_2$-type.