Let $K$ be a function field of odd characteristic, and let $\pi$
(resp., $\eta$) be a cuspidal automorphic representation of ${\rm
GL}\sb 2(\mathbb {A}\sb K)$ (resp., ${\rm GL}\sb 1(\mathbb {A}\sb
K)$). Then we show that a weighted sum of the twists of $L(s,\pi)$ by
quadratic characters $\chi\sb D,\sum \sb DL(s,\pi\otimes \sp \chi\sb
D)a\sb 0(s,\pi,D)\eta(D)|D|\sp {-w}$, is a rational function and has a
finite, nonabelian group of functional equations. A similar
construction in the noncuspidal cases gives a rational function of
three variables. We specify the possible denominators and the degrees
of the numerators of these rational functions. By rewriting this
object as a multiple Dirichlet series, we also give a new description
of the weight functions $a\sb 0(s,\pi,D)$ originally considered by
D. Bump, S. Friedberg and J. Hoffstein.