On the functional equation of the standard twist associated with half-integral weight cusp forms

Let $F(s)$ be the normalized Hecke $L$-function associated with a cusp form of half-integral weight $\kappa$ and level $N$. We show that the standard twist $F(s,\alpha)$ of $F(s)$ satisfies a functional equation reflecting $s$ to $1-s$. The shape of the functional equation is not far from a standard Riemann type functional equation of degree 2; actually, it may be regarded as a degree 2 analog of the Hurwitz-Lerch functional equation. We also deduce some result on the order of growth on vertical strips and on the distribution of zeros of $F(s,\alpha)$.