The non-projective part of the tensor powers of a module

Let $M$ be a finite dimensional modular representation of a finite group $G$. We consider the generating function for the non-projective part of the tensor powers of $M$, and we write $\gamma_G(M)$ for the reciprocal of the radius of convergence of this power series. We investigate the properties of the invariant $\gamma_G(M)$, using tools from representation theory, and from the theory of commutative Banach algebras.