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Deligne has formulated extremely influential conjectures about certainspecial values of the $L$-functions of (Grothendieck) motives over a numberfield $F$. Given the conjectural dictionary between motives and 'algebraic'automorphic representations of $\textrm{GL}(N, {\mathbb A}_F)$, where ${\mathbbA}_F$ denotes the ad\`eles of $F$, they translate into conjectures concerningthe $L$-functions of these automorphic representations. These complexrepresentations, when they are 'regular', can be conjugated by theautomorphisms of the complex field ${\mathbb C}$. It then follows, as a weakconsequence of Deligne's conjectures, that the vanishing at critical points(integers of half-integers) of the automorphic $L$-functions should beinvariant by automorphisms of ${\mathbb C}$. If $F$ is totally imaginary, thishas been proven by Moeglin, for standard or Rankin $L$-functions. Here weextend the result to Rankin $L$-fuctions for totally real fields $F$, under aparity and a regularity assumption. The proof relies on Eisenstein cohomologyand the Zucker conjecture (a theorem of Looijenga and Saper-Stern.)

On the central value of Rankin $L$-functions for self-dual algebraic representations of linear groups over totally real fields