Suppose that we have a Zariski dense subgroup $\Gamma$ of $GL(d,\mathbb{R})$. Let $\delta$ be the abscissa of convergence of the series $$ \sum_{x \in \Gamma} e^{-s \|x\|} $$ and suppose that this series diverges at $s = \delta$. If we have a subsemigroup $N$ of $\Gamma$ such that $$ \sum_{x \in N} e^{-\delta \|x\|} = \infty $$ can I conclude that $N$ is Zariski dense in $GL(d,\mathbb{R})$?

Intuitively this feels like it should be true - but I can't see a way in to prove it! Any suggestions would be appreciated.