  Given $k\in\mathbb N$, we study the vanishing of the Dirichlet series
$$D_k(s,f):=\sum_{n\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a
periodic function modulo a prime $p$. We show that if $(k,p-1)=1$ or
$(k,p-1)=2$ and $p\equiv 3\mod 4$, then there are no odd rational-valued
functions $f\not\equiv 0$ such that $D_k(1,f)=0$, whereas in all other cases
there are examples of odd functions $f$ such that $D_k(1,f)=0$.
  As a consequence, we obtain, for example, that the set of values
$L(1,\chi)^2$, where $\chi$ ranges over odd characters mod $p$, are linearly
independent over $\mathbb Q$.
