Algebraicity of ratios of special $L$-values for $\mathrm{G}\mathrm{L}(n)$

We prove, under certain assumptions, algebraicity of the ratio $L(m,\mathrm{\Pi }\times \chi )\mathrm{/}L(m,\mathrm{\Pi }\times {\chi }^{\mathrm{\prime }})$, where $\mathrm{\Pi }$ is a cuspidal automorphic cohomological unitary representation of ${\mathrm{G}\mathrm{L}}_n({\mathbb{A}}_{\mathbb{Q}})$, and $\chi$, ${\chi }^{\mathrm{\prime }}$ are finite order Hecke characters such that ${\chi }_{\mathrm{\infty }}={\chi }_{\mathrm{\infty }}^{\mathrm{\prime }}={\mathrm{s}\mathrm{g}\mathrm{n}}^r$, and $m,r$ are specific positive integers which depends only on ${\mathrm{\Pi }}_{\mathrm{\infty }}$. The methods in this article are a generalization of those in the work of Mahnkopf [Cohomology of arithmetic groups, parabolic subgroups and the special values of $L$-functions of GL(n), J. Inst. Math. Jussieu, 4 (2005)].