On the non-vanishing of certain Dirichlet series

Given k∈N, we study the vanishing of the Dirichlet series. Dk(s,f):=∑n≥1dk(n)f(n)n-s at the point s=1, where f is a periodic function modulo a prime p. We show that if (k,p-1)=1 or if (k,p-1)=2 and p≡3 (mod 4), then there are no odd rational-valued functions f≢0 such that Dk(1,f)=0, whereas in all other cases there are examples of odd functions f such that Dk(1,f)=0.As a consequence, we obtain, for example, that the set of values L(1,χ)2, where χ ranges over odd characters mod p, are linearly independent over Q.