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