For all but finitely many anticyclotomic twists of an equivariant Hecke character over any imaginary quadratic field, the central vanishing order of the L-function is 0 or 1 exactly according to the root number.
Iwaniec, Topics in Classical Automorphic Forms , Graduate Studies in Mathematics, Vol
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On $L$-functions of Hecke characters and anticyclotomic towers
For all but finitely many anticyclotomic twists of an equivariant Hecke character over any imaginary quadratic field, the central vanishing order of the L-function is 0 or 1 exactly according to the root number.