  Let $\omega$ be a weight function defined on a locally compact group $G$,
$1\le p<+\infty$, $S\subset G$ and let us assume that for any $s\in S$, the
left translation operator $T_s$ is continuous from the weighted $L^p$-space
$L^p(G,\omega)$ into itself. For a given set $\Gamma\subset\mathbb{C}$, a
vector $f\in L^p(G,\omega)$ is said to be $(\Gamma,S)$-dense if the set $\{
\lambda T_sf:\, \lambda\in \Gamma, \,s\in S\}$ is dense in $L^p(G,\omega)$. In
this paper, we characterize the existence of $(\Gamma,S)$-dense vectors in
$L^p(G,\omega)$ in terms of the weight and the set $\Gamma$.
