The BSD conjecture gives an interpretation to the order of vanishing of the central value the L-function of an elliptic curve (it is supposed to be the rank of that elliptic curve), while the modularity theorem says that any such L-function is actually the L-function of some Hecke modular form.
Has there been work on the meaning of the order of vanishing of a central value of the L-function of a Hecke eigenform in this other context (other than the straightforward ones about the magnitude of the coefficients)?