  We show that every Fricke invariant meromorphic modular form for
$\Gamma_0(N)$ whose divisor on $X_0(N)$ is defined over $\mathbb{Q}$ and
supported on Heegner divisors and the cusps is a generalized Borcherds product
associated to a harmonic Maass form of weight $1/2$. Further, we derive a
criterion for the finiteness of the multiplier systems of generalized Borcherds
products in terms of the vanishing of the central derivatives of $L$-function
of certain weight $2$ newforms. We also prove similar results for twisted
Borcherds products.
