  In this paper, we prove that if the Fourier coefficients of a
$\mathrm{SL}(3,\mathbb{Z})$ Hecke--Maa\ss\ cusp form $\pi$ are not too
correlated with additive characters, then there exists infinitely many
Dirichlet characters such that \begin{align*}
  L\left(\frac{1}{2},\pi\otimes\chi\right)L\left(\frac{1}{2},\chi\right)\neq 0.
\end{align*} To prove this result, we compute the first twisted moment of these
$L$ function averaged over a well chosen set of conductors.
