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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 toocorrelated with additive characters, then there exists infinitely manyDirichlet 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.

Non vanishing of products of twisted $\mathrm{GL}(3)$ $L$-functions