Do Artin L functions have polynomial growth in in the critical strip? Given an irreducible representation $\rho$ of the Galois group $G$ of a number field $K$ over $\mathbb{Q}$, we have the associated Artin $L$ function which we denote by $L(s, \rho)$. By Brauer induction (along with Artin reciprocity and the work of Hecke), $L(s, \rho)$ is known to have a meromorphic continuation to the whole complex plane. The Artin holomorphy conjecture states that $L(s, \rho)$ is analytic if $\rho$ is not the trivial representation.
My question is regarding the growth of $L(s, \rho)$ in the critical strip $0 < Re(s) < 1 $ as $Im(s) \to \infty$. My intuition tells me that $L(s,\rho)$ should have polynomial growth in $Im(s)$ as $Im(s)\to\infty$ (and indeed it is if we assume automorphy). But I feel that assuming automorhphy is way too big a prize to pay for something that should be much easily available.
Any help would be greatly appreciated.
 A: By Artin, Brauer, and Hecke, there exist integers $c_{\chi,\rho}$ such that
$$L(s,\rho) = \prod_{H\subseteq G} \prod_{\substack{\text{$\chi$ an irrep. of $H$} \\ \text{$\dim\chi=1$}}} L(s,\chi)^{c_{\chi,\rho}}.$$
Each $L(s,\chi)$ is entire, but unless all of the $c_{\chi,\rho}$ are shown to be nonnegative, standard results on the distribution of zeros of $L(s,\chi)$ show that we cannot preclude the possibility that there exist poles of $L(s,\rho)$ at points $z$ with $0<\mathrm{Re}(z)<1$ and $|\mathrm{Im}(z)|$ arbitrarily large.  As such, one must be careful when one says "$\mathrm{Im}(s)\to\infty$", and one must restrict $\mathrm{Im}(s)$ to certain sequences.  If two poles $z_1$ and $z_2$ satisfy $|z_1-z_2|<1$, then basic results on the distribution of zeros of $L(s,\chi)$ show that $|z_1-z_2|\gg \frac{1}{\log|\mathrm{Im}(z_1)|}$.  So there exist many sequences of complex numbers $s$ along which we attain at most polynomial growth of $L(s,\rho)$ in the $|\mathrm{Im}(s)|$-aspect.
Alternatively, one could restrict the real part of $s$ to subsets of the critical strip where $L(s,\rho)$ is free of poles.  Such a region exists because of known zero-free regions for the $L(s,\chi)$.  For example, there exists a constant $c=c(G,K)>0$ such that $L(\sigma+it,\rho)$ grows polynomially in $t$ as long as $\sigma\geq 1-c/\log(|t|)$.
In order to say much more, one must handle the pole distribution.  Without something like an assumption (or proof) of automorphy, we cannot say very much apart from special cases.  The most notable such special case was given by Aramata and Brauer:  If $K/F$ is a Galois extension of number fields, then $L(s,\rho) = \zeta_K(s) / \zeta_F(s)$ is entire (in particular, the $c_{\chi,\rho}$ are all nonnegative).  In this case, the polynomial growth is immediate from the polynomial growth of the $L(s,\chi)$.
