  The analytic properties of the standard twist $F(s,\alpha)$, where $F(s)$
belongs to a wide class of $L$-functions, are of prime importance in describing
the structure of the Selberg class. In this paper we present a deeper study of
such properties. In particular, we show that $F(s,\alpha)$ satisfies a
functional equation of a new type, somewhat resembling that of the
Hurwitz-Lerch zeta function. Moreover, we detect the finer polar structure of
$F(s,\alpha)$, characterizing in two different ways the occurrence of finitely
or infinitely many poles as well as giving a formula for their residues.
