Let $G$ be a locally compact group and $1\le p<\infty$. In this paper, we show that the mapping $\pi: G\times L^p(G)\to L^p(G)$ with the assignment $(t, f)\mapsto T_t f=\frac{\phi}{\delta_t*\phi}\delta_t*f$, is separately continuous which ensures that $\{T_t\}_{t\in G}$ forms a $C_0$-semigroup. Also, it is proved that the $C_0$-semigroup $\{T_t\}_{t\in G}$ admits a universal vector only when $G$ is noncompact and second countable. %In the case of the existence of universal vectors for $\{T_t\}_{t\in G}$, we %approximate its orbit by the scalar multiplication of the weight %function on any compact subset of $G$. The hypercyclic phenomenon is studied for the single weighted translation as well. Finally, we give examples of universal $C_0$-semigroups.
