TITLE: A natural way of thinking of the definition of an Artin $L$-function?
QUESTION [16 upvotes]: Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be $\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 - N(\mathfrak{p})^{-s}}$. He also knew that if $L/K$ is a class field then $\displaystyle\prod_{\mathfrak{P}|\mathfrak{p}}\frac{1}{1 - N(\mathfrak{P})^{-s}} = \displaystyle\prod_{\chi}\frac{1}{1 - \chi{(Frob{_\mathfrak{p})}}\cdot N(\mathfrak{p})^{-s}}$ where $\mathfrak{P}$ runs over all primes in $L$ lying above $\mathfrak{p}$ and $\chi$ runs over all characters of $Gal(L/K)$. 
It's natural then to 

Define $L$-series attached to characters on $Gal(L/K)$.
Recognize that the definition makes sense whether or not $L/K$ is a class field.
In light of the fact that characters are $1$-dimensional representations of $Gal(L/K)$, ask whether there's a good definition of the $L$-series attached to a higher dimensional representation of non-abelian $Gal(L/K)$. 

But having come this far, how does one then arrive at the definition of the local factor of an $L$-series attached to a representation $\rho: Gal(L/K) \to GL_{n}(\mathbb{C})$ at a prime $\mathfrak{p}$ unramified in $K$  as 
$\displaystyle \frac{1}{\det(Id - \rho(Frob_\mathfrak{p})N(\mathfrak{p})^{-s})}$
?
To be sure

It specializes to the definition of the $L$-series attached to a character on $Gal(L/K)$.
It's well-defined (independent of which member of the conjugacy class $Frob_\mathfrak{p}$ one chooses).
One has the theorem $\zeta_{L/\mathbb{Q}} = \prod_{\rho} L(\rho, s)$ where $\rho$ ranges over irreducible representations of $Gal(L/\mathbb{Q})$, generalizing the analogous fact for characters on Galois groups of class fields.

And perhaps the three properties listed above are sufficient to uniquely determine the definition. (Maybe one needs more than the above three, I would have to think about it it.) Maybe this is how Artin discovered the definition. This line of thinking is similar to Feynman's heuristic derivation of Heron's formula. But I somehow feel as though this doesn't get at the essence of things. Is there a way of thinking about the definition of an Artin L-series that gives it more of a sense of inevitability and canonicity?
[Reposted from mathstackexchange.]

REPLY [2 votes]: Since this is near the top again, I'll add what seems natural to me. Let $F/\mathbb{Q}$ be an $S_3$ extension, and let $C$ and $K$ be the cubic and quadratic subfields. Then the $\zeta$ functions of these number fields have Euler products.
Let $p \in \mathbb{Z}$ be a prime which is not ramified in $F$. Then the corresponding Euler factor in each of the fields $\mathbb{Q}$, $K$, $C$ and $F$ is dictated by whether the Frobenius element has conjugacy class $e$, $(12)$ or $(123)$. Here is the formula for the Euler factor in each case:
$$\begin{array}{|c|c|c|c|}
\hline
& e & (12) & (123) \\
\hline
\zeta_\mathbb{Q} & (1-p^{-s})^{-1} & (1-p^{-s})^{-1} & (1-p^{-s})^{-1} \\
\hline
\zeta_K & (1-p^{-s})^{-2} & (1-p^{-s})^{-1}(1+p^{-s})^{-1}&(1-p^{-s})^{-2} \\
\hline
\zeta_C & (1-p^{-s})^{-3} & (1-p^{-s})^{-2} (1+p^{-s})^{-1}&(1-p^{-s})^{-1}(1+p^{-s}+p^{-2s})^{-1} \\
\hline
\zeta_F & (1-p^{-s})^{-6} &(1-p^{-s})^{-3}(1+p^{-s})^{-3}& (1-p^{-s})^{-2}(1+p^{-s}+p^{-2s})^{-2}\\
\hline
\end{array}$$
Looking at this table makes it natural to imagine additional rows for $L$-functions $L_1$ and $L_2$ with $\zeta_K = \zeta_{\mathbb{Q}} L_1$,  $\zeta_C = \zeta_{\mathbb{Q}} L_2$ and  $\zeta_F = \zeta_{\mathbb{Q}} L_1 L_2^2$ and Euler factors
$$\begin{array}{|c|c|c|c|}
\hline
& e & (12) & (123) \\
\hline
L_1 & (1-p^{-s})^{-1} &(1+p^{-s})^{-1} &(1-p^{-s})^{-1}  \\
\hline
L_2 &(1-p^{-s})^{-2} &(1-p^{-s})^{-1} (1+p^{-s})^{-1}& (1+p^{-s}+p^{-2s})^{-1}\\
\hline
\end{array}$$
(It especially helps that Hecke and Dirichlet already studied $L_1$.)
We now have three things associated to $S_3$, namely $\zeta_{\mathbb{Q}}$, $L_1$ and $L_2$, and more complicated things are made from them. In particular, $\zeta_F$ has 1 copy of the "trivial" $\zeta_{\mathbb{Q}}$, 1 copy of $L_1$ and $2$ copies of $L_2$. That already screams representation theory to me. Then when I notice that the degree $d$ representations gives degree $d$ polynomials in $p^{-s}$ in the denominator, what could be more natural than a characteristic polynomial?