  We address the question of non-vanishing of $L(1,f)$ where $f$ is an
algebraic-valued, periodic arithmetical function. We do this by characterizing
algebraic-valued, periodic functions $f$ for which $L(1,f)=0$. The case of odd
functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result
of Bass to obtain a characterization for the even functions. We also describe a
theorem of the first two authors which says that it is enough to consider only
the even and the odd functions in order to obtain a complete characterization.
