  A systematic study is presented for centrality, transverse momentum ($p_T$)
and pseudorapidity ($\eta$) dependence of the inclusive charged hadron elliptic
flow ($v_2$) at midrapidity($|\eta| < 1.0$) in Au+Au collisions at
$\sqrt{s_{NN}}$ = 7.7, 11.5, 19.6, 27 and 39 GeV. The results obtained with
different methods, including correlations with the event plane reconstructed in
a region separated by a large pseudorapidity gap and 4-particle cumulants
($v_2{4}$), are presented in order to investigate non-flow correlations and
$v_2$ fluctuations. We observe that the difference between $v_2{2}$ and
$v_2{4}$ is smaller at the lower collision energies. Values of $v_2$, scaled by
the initial coordinate space eccentricity, $v_{2}/\varepsilon$, as a function
of $p_T$ are larger in more central collisions, suggesting stronger collective
flow develops in more central collisions, similar to the results at higher
collision energies. These results are compared to measurements at higher
energies at the Relativistic Heavy Ion Collider ($\sqrt{s_{NN}}$ = 62.4 and 200
GeV) and at the Large Hadron Collider (Pb + Pb collisions at $\sqrt{s_{NN}}$ =
2.76 TeV). The $v_2(p_T)$ values for fixed $p_T$ rise with increasing collision
energy within the $p_T$ range studied ($< 2 {\rm GeV}/c$). A comparison to
viscous hydrodynamic simulations is made to potentially help understand the
energy dependence of $v_{2}(p_{T})$. We also compare the $v_2$ results to UrQMD
and AMPT transport model calculations, and physics implications on the
dominance of partonic versus hadronic phases in the system created at Beam
Energy Scan (BES) energies are discussed.
