  We derive a precise relation of poles of Eisenstein series associated to the
cuspidal datum $\chi\otimes\sigma$ and lowest occurrence of theta lifts of a
cuspidal automorphic representation $\sigma$ of a unitary group, where $\chi$
is conjugate self-dual character. We also give a refined result on
non-vanishing of periods of Eisenstein series and first occurrence of theta
lifts. This gives constraints on existence of $(\chi,b)$-factors in the global
$A$-parameter of $\sigma$.
