langlands conjectures - Automorphic representations attached to abelian varieties

Let $A$ be an abelian variety defined over $mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $pi_A$ of $GL(2d)/mathbb{Q}$ whose L-function agrees with the L-function attached to the Tate module of $A$. In fact, $pi_A$ should arise from an automorphic form on an orthogonal group. My question is, which (real) form of $O(2d+1)$ should $pi_A$ live on? For $d=1$ the group is $O(2,1)$, which is isogenous to $SL_2$, and for $d=2$ the group should be is $O(2,3)$, which is isogenous to $Sp_4$. But what should happen in general?