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The purpose of this paper is to study the local zeta integrals ofFriedberg-Jacquet at complex place and to establish similar results to ourrecent work in the reall case joint with C. Cheng and D. Jiang. In this paper,we will (1) give a necessary and sufficient condition on an irreducibleessentially tempered cohomological representation $\pi$ of$\mathrm{GL}_{2n}(\mathbb{C})$ with a non-zero Shalika model; (2) construct anew twisted linear period $\Lambda_{s,\chi}$; (3) give a necessary andsufficient condition on the character $\chi$ such that there exists a uniformcohomological test vector $v\in V_\pi$ (which we construct explicitly) for$\Lambda_{s,\chi}$. As a consequence, we obtain the non-vanishing of localFriedberg-Jacquet integral at complex place. All of the above are essentialpreparations for attacking a global period relation problem.

Archimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of $\mathrm{GL}_{2n}$: Complex Case