Let $\pi_p$ be a ramified representation of $GL(n,\mathbb{Q}_p)$. Let $\chi_p$ be a ramified representation of $GL(1,\mathbb{Q}_p)$.

Is it generally known that $L(s,\pi_p\times \chi_p)=1$ if $\chi_p$ is ramified enough (higher conductor)?

I believe this is known for $n=2$ or even general $n$. Can you give some reference?