We consider certain families of Hecke characters ϕ over a quadratic imaginary field F. The order of vanishing of the L-function L(ϕ,s) at the central point s=−1, according to the Beilinson conjectures, should be equal to the dimension of the space of extensions of the Tate motive Q(1) by the motive associated with ϕ. In this article, candidates for the corresponding extensions of Hodge structure are constructed, under the assumptions that the sign of the functional equation of L(ϕ,s) is −1 and that L′(ϕ,−1)≠0. This is achieved by means of the cohomology of variations of Hodge structures over Picard modular surfaces attached to F and Harder's theory of Eisenstein cohomology. Moreover, we provide a criterion for the non-triviality of these extensions, based on the non-vanishing of a height pairing that we define and study.