Tate classes and L-functions on a product of a quaternionic Shimura surface and a Picard modular surface

We compute the space of Tate classes on a product of a quaternionic Shimura surface and a Picard modular surface in terms of automorphic representations including the exact determination of their field of definition and prove the equality between the dimension of the space of Tate classes and the order of the pole at s=3 of the L-function in some special cases.