In this paper, we propose $λ_{g}$ conjecture for Hodge integrals with target varieties. Then we establish relations between Virasoro conjecture and $λ_{g}$ conjecture, in particular, we prove $λ_{g}$ conjecture in all genus for smooth projective varieties with semisimple quantum cohomology or smooth algebraic curves. Meanwhile, we also prove $λ_{g}$ conjecture in genus zero for any smooth projective varieties. In the end, together with DR formula for $λ_g$ class, we obtain a new type of universal constraints for descendant Gromov-Witten invariants. As an application, we prove $λ_{g}$ conjecture in genus one for any smooth projective varieties.