Assume $f$ is a homogeneous integer polynomial in $n\geq 3$ variables such that the hypersurface $f=0$ is irreducible over $\mathbb{Q}$ (but not necessarily over $\overline{\mathbb{Q}}$ so for example we are allowed to consider $f=x_0^2+x_1^2$).
The Hasse-Weil zeta function $\zeta_f(s), s \in \mathbb{C},$ describes the number of solutions of the hypersurface $f=0$ collectively over all primes $p$. I would like to ask whether in this generality it is known that the real point on the abscissa of convergence has a pole of order one.
This is true if one has $2$ variables since this would follow from Landau's prime number theorem for number fields. Also if the hypersurface is smooth over $\mathbb{Q}$ then it should also be true. However the interesting case regards hypersurfaces that are irreducible over $\mathbb{Q}$ and singular over $\overline{\mathbb{Q}}$, one such example being $$f=x_0x_1x_2-x_3^3.$$