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In this paper, we show nonvanishing of some Hecke characters on cyclotomicfields. The main ingredient of this paper is a computation of eigenfunctionsand the action of Weil representation at some primes including the primes above$2$. As an application, we show that for each isogeny factor of the Jacobian ofthe $p$-th Fermat curve where $2$ is a quadratic residue modulo $p$, there areinfinitely many twists whose analytic rank is zero. Also, for a certainhyperelliptic curve over the $11$-th cyclotomic field whose Jacobian hascomplex multiplication, there are infinitely many twists whose analytic rank iszero.

Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields