On L-functions of quadratic ℚ-curves

Let K be a quadratic number field and let E be a ℚ-curve without CM completely defined over K and not isogenous to an elliptic curve over ℚ. In this setting, it is known that there exists a weight 2 newform of suitable level and character, such that L(E, s) = L(f, s)L(σf, s), where σf is the unique Galois conjugate of f. In this paper, we first describe an algorithm to compute the level, the character and the Fourier coefficients of f. Next, we show that given an invariant differential ωE on E, there exists a positive integer Q = Q(E, ωE) such that L(E, 1)/P(E/K) · Q is an integer, where P(E/K) is the period of E. Assuming a generalization of Manin's conjecture, the integer Q is made effective. As an application, we verify the weak BSD conjecture for some curves of rank two, we compute the L-ratio of a curve of rank zero and we produce relevant examples of newforms of large level.