  The purpose of this paper is to study the local zeta integrals of
Friedberg-Jacquet at complex place and to establish similar results to our
recent 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 irreducible
essentially tempered cohomological representation $\pi$ of
$\mathrm{GL}_{2n}(\mathbb{C})$ with a non-zero Shalika model; (2) construct a
new twisted linear period $\Lambda_{s,\chi}$; (3) give a necessary and
sufficient condition on the character $\chi$ such that there exists a uniform
cohomological test vector $v\in V_\pi$ (which we construct explicitly) for
$\Lambda_{s,\chi}$. As a consequence, we obtain the non-vanishing of local
Friedberg-Jacquet integral at complex place. All of the above are essential
preparations for attacking a global period relation problem.
