  We consider a certain class of multiplicative functions $f: \mathbb N
\rightarrow \mathbb C$ and study the distribution of zeros of Dirichlet
polynomials $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ corresponding to these
functions. We prove that the known non-trivial zero-free half plane for
Dirichlet polynomials associated to this class of multiplicative functions is
optimal. We also introduce a characterization of elements in this class based
on a new parameter depending on the Dirichlet series $F(s) = \sum_{n=1}^\infty
f(n) n^{-s}$. In this context, we obtain non-trivial regions in which the
associated Dirichlet polynomials do have zeros.
