  Let $G$ be the $ax+b$ group. There are essentially two irreducible infinite
dimensional unitary representations of $G$, $(\mu, L^2(\mathbb R^+))$ and
$(\mu^*, L^2(\mathbb R^+))$. In this paper, we give various characterizations
about smooth vectors of $\mu$ and their Mellin transforms. Let $\f d$ be a
linear sum of delta distributions supported on the the positive integers
$\mathbb Z^+$. We study the Mellin transform of the matrix coefficients $\mu_{
\f d, f}(a)$ with $f$ smooth. We express these Mellin transforms in terms of
the Dirichlet series $L(s, \f d)$. We determine a sufficient condition such
that the generalized matrix coefficient $\mu_{\f d, f}$ is a locally integrable
function and estimate the $L^2$-norms of $\mu_{\f d, f}$ over the Siegel set.
We further derive an inequality which may potentially be used to study the
Dirichlet series $L(s, \f d)$.
