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It is proved that the Hasse-Weil zeta functions of the canonical componentsof the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientablecomplete hyperbolic $3$-manifolds of finite volume are equal to the Dedekindzeta functions of their trace fields (invariant trace fields). When the closed$3$-manifold is arithmetic, the special value at $s=2$ of the Hasse-Weil zetafunction of the canonical component of the ${\rm PSL}_2$-character variety isexpressed in terms of the hyperbolic volume of the manifold up to rationalnumbers.

Hasse-Weil zeta functions of ${\rm SL}_2$-character varieties of closed orientable hyperbolic $3$-manifolds