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We evaluate asymptotically the smoothed first moment of central values offamilies of quadratic, cubic, quartic and sextic Hecke $L$-functions overvarious imaginary quadratic number fields of class number one, using the methodof double Dirichlet series. In particular, we obtain asymptotic formulas forthe quadratic families with error terms of size $O(X^{1/4+\varepsilon})$ underthe generalized Riemann hypothesis.

First moment of central values of Hecke $L$-functions with Fixed Order Characters