  Let $\alpha>1$ be an irrational number of finite type $\tau$. In this paper,
we introduce and study a zeta function $Z_\alpha^\sharp(r,q;s)$ that is closely
related to the Lipschitz-Lerch zeta function and is naturally associated with
the Beatty sequence ${\mathcal B}(\alpha):=(\lfloor\alpha
m\rfloor)_{m\in{\mathbb N}}$. If $r$ is an element of the lattice ${\mathbb
Z}+{\mathbb Z}\alpha^{-1}$, then $Z_\alpha^\sharp(r,q;s)$ continues
analytically to the half-plane $\{\sigma>-1/\tau\}$ with its only singularity
being a simple pole at $s=1$. If $r\not\in{\mathbb Z}+{\mathbb Z}\alpha^{-1}$,
then $Z_\alpha^\sharp(r,q;s)$ extends analytically to the half-plane
$\{\sigma>1-1/(2\tau^2)\}$ and has no singularity in that region.
