On mazur's conjecture for twisted L-functions of elliptic curves over totally real or CM fields

Let E be an elliptic curve defined over a number field F, and let Σ be a finite set of finite places of F. Let L(s, E, ψ) be the L-function of E twisted by a finite-order Hecke character ψ of F. It is conjectured that L(s, E, ψ) has a meromorphic continuation to the entire complex plane and satisfies a functional equation s ↔ 2 s. Then one can define the so called minimal order of vanishing at s = 1 of L(s, E, ψ), denoted by m(E, ψ) (see Section 2 for the definition).