As to the meaning of the symbols, the reader is referred to the introduction of Part I. We recall here only that F is a totally real algebraic number field of degree n, and w denotes the Fourier coefficients of an elliptic modular form 52(z) = E wQ(a)e2q'iaz. In Part I, we investigated D assuming that EX(a)N(Q) S is an L-function of a CM-field with an algebraic-valued Hecke character. In the present Part II, we treat the case where X( a) can be obtained as eigenvalues of Hecke operators T(a) on a quaternion algebra B over F. Thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane & with respect to congruence subgroups of B x . Here r is the number of archimedean primes of F unramified in B. Then we introduce in Section 2 the notion of arithmeticity (or Q-rationality) of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones. Our main theorems will be stated in Section 3 and proved in Section 6; Sections 4 and 5 are preliminaries to the proof. The central result, specialized to the case r = 1, asserts that if g is a Q-rational eigenform such that gj T( a) =X( a )g for all integral ideals a of F, then D(s0) for certain integers so are algebraic numbers times 7Tdg, g), where d is an integer determined by the weight of g, and (*, *) is the Petersson inner product.