Positive Definiteness of a Family of Functions

General necessary conditions on the real parameters $\alpha$, $\beta$, $C$, $D$ for the function
$$ e^{-\alpha\rho(x)}(C\cos\beta\rho(x)+D\sin\beta\rho(x)), $$
where $\rho$ is the norm on $\mathbb R^n$, to be positive definite on $\mathbb R^n$, are obtained. For $\rho(x)=\|x\|_p$, a criterion on these parameters is obtained in the following cases: (i) $p=1$ or $p=2$; (ii) $3<p\le\infty$ and $n=2$.