  Given a branching random walk$\{Z_n\}_{n\geq0}$ on $\mathbb{R}$, let
$Z_n([y,\infty))$ be the number of particles located in $[y,\infty)$ at
generation $n$. It is known from \cite{Biggins1977} that under some mild
conditions, $n^{-1}\log Z_n([\theta x^* n,\infty))$ converges a.s. to $\log
m-I(\theta x^*)$, where $\log m-I(\theta x^*)$ is a positive constant. In this
work, we investigate its lower deviation, in other words, the convergence rates
of $$\mathbb{P}\left(Z_n([\theta x^* n,\infty))<e^{an}\right),$$ where
$a\in[0,\log m-I(\theta x^*))$. Our results complete those in \cite{Mehmet},
\cite{Helower} and \cite{GWlower}.
