Suppose $\rho:G _{\mathbb{Q}} \rightarrow GL_n(\mathbb{Q}_p)$ is a Galois rep. It has a uniquely defined (up to semisimplification residual rep $\bar{\rho}$. $\bar{\rho}$ is unramified where $\rho$ is. I have examples where it becomes good at a prime where $\rho$ is bad. Can it become more ramified then $\rho$? What constraints on the degree of the ramification of $\bar{\rho}$ do we know in terms of ramification of $\rho$.

Suppose now we have an $L$ function, and know the bad factors of the $L$ function. Does this help us further? What if $p$ is itself a bad prime: can we still say anything? Note that I cannot assume that this Galois rep is motivic.