Conventions: sheaf = complex of constructible sheaves (in the l-adic setup with etale tplg or in the complex coefficients setup with analytical tplg).

I would like to understand if there is an intuition behind the following property of a sheaf. We consider $X$ a variety (or a scheme, or a gentle alg. stack) not necessarily smooth/proper. What sort of sheaves $\mathcal{F}$ on $X$ satisfy the following property: $Rp_!(\mathcal{F}) = 0$ where $p:X\to \{*\}$. Or (by Verdier duality) the property:

$Rp_*(\mathcal{F}) = 0$.

Any (partial) answer in any particular case (say X smooth, or X proper and smooth, etc.) would be nice.

------- motivation for the question -----

I encountered the notion of "regular sheaf" in the paper of Braverman & Gaitsgory - "Geometric Eisenstein series" which uses the above property. In their context the map $p$ is from $Bun_{T}\to Bun_{T/G_m}$, where $T$ is a torus in a reductive group ($Bun_T$ is over a smooth proj curve); the map $p$ is induced from a cocharacter $G_m\to T$. They call a sheaf regular if $Rp_!\mathcal{F} = 0$ for all the maps $p$ as above that are induced from a coroot. This notion seems to be very important in their paper and I'm trying to understand intuitively this condition and see some simple examples.