On the generalized Hua problem

We determine the precise value of the exponent of convergence of the improper integral
$$ \gamma_1=\int _{\mathbb R^r}\biggl|\int_0^1e^{2\pi if(x)}\,dx\biggr|\,d\alpha_1\dots d\alpha_r, $$
where $f(x)=\alpha_1x^{C_1}+\dots+\alpha_rx^{C_r}$, $0<C_1<\dots<C_r$ are arbitrary real numbers.