Constraints on $z$ such that $z r_1, \ldots, z r_n$ still have the same minimal polymial

Question

Let $\mathbb{Q}$ be the base field, and $r_1, \ldots, r_n$ be all the roots of a minimal polynomial $p \in \mathbb{Q}[x]$ of rank $n$. Suppose I multiply each of these roots by the same $z \in \mathbb{C}$. What would the conditions on $z$ be such that all new roots $z r_1, \ldots, z r_n$ have the same minimal polynomial $q\in\mathbb{Q}[x]$? $q$ is allowed to have a different rank from $p$.