New measurements of directed flow for charged hadrons, characterized by the Fourier coefficient v1 , are presented for transverse momenta pT , and centrality intervals in Au+Au collisions recorded by the STAR experiment for the center-of-mass energy range sNN=7.7–200 GeV. The measurements underscore the importance of momentum conservation, and the characteristic dependencies on sNN , centrality and pT are consistent with the expectations of geometric fluctuations generated in the initial stages of the collision, acting in concert with a hydrodynamic-like expansion. The centrality and pT dependencies of v1even , as well as an observed similarity between its excitation function and that for v3 , could serve as constraints for initial-state models. The v1even excitation function could also provide an important supplement to the flow measurements employed for precision extraction of the temperature dependence of the specific shear viscosity.