Dear Barinder,
Re. your comment "there cannot be a common generalization of Artin and Hecke $L$-series",
to the contrary, there is such a common generalization, namely the $L$-series of a representation of the global Weil group. These will (conjecturally) have an analytic continuation and functional equation, and they include all Hecke $L$-series (Hecke characters, by which I mean idele class characters, are just one-dimensional reps. of the global Weil group), and all Artin $L$-series (which are reps. of the global Weil group which factor
through the map to $G_K$).
Regards,
Matthew
Artin's work on zeta functions began in 1923 (actually zeta
functions had already played a role in his thesis on quadratic
extensions of the rational function field) with an article
"On the zeta functions of certain algebraic number fields".
There he studied a problem due to Dedekind which asked whether
the zeta function of a number field is always divisible (in the
sense that the quotient is entire) by the zeta function of any
of its subfields. Dedekind had proved this for purely cubic
fields, and for abelian (and in fact metabelian, then called
metacyclic) extensions it follows from the decomposition of
Dedekind's zeta functions into a product of abelian L-series
due to Takagi's class field theory.
Artin then computed explicitly the zeta functions for subfields of
an $S_4$-extension, where the factors contributed by a prime ideal
${\mathfrak p}$ depends on the decomposition group of ${\mathfrak p}$,
and then he sketched a similar calculation for the icosahedral group.
For unramified primes, these factors all have a natural interpretation
in terms of the Frobenius automorphisms, or, in other words, come from
a Galois representation. One can do worse than read Harold Stark's
beautiful article in the book "From number theory to physics", where
even simpler examples all presented in all their glory.
In Artin's first article on L-series (On a new kind of L-series, 1923)
Artin defined the Euler factors of the L-series attached to a Galois
representation only for unramified primes. This was sufficient (if not
very satisfying) for the following reason: Artin could write the
zeta function of $K$ and all of its subfields as products of his L-series.
Hecke had shown in 1917 that L-series whose Euler factors agree up to at
most finitely many primes actually have equal Euler factors if both L-series
satisfy the same functional equation. So if you can show that Artin's
L-series satisfy a functional equation with suitably defined (but not
explicitly known) factors at the ramified and infinite primes, then
everything is fine. At the end of this artice, Artin takes up his
example of the icosahedral group again.
In his sequel "On the theory of L-series with general group characters"
from 1930, Artin observed that the state of the theory was not
satisfactory and proceeded to define the "local" factors (local
class field theory was being developed simultaneously by Hasse;
Artin's reciprocity law had allowed a new approach to the norm
residue symbols, and this led more or less automatically to local
class field theory) from the start. He does this by starting with
a Galois representation, observing that for ramified primes, the
"Frobenius automorphism" is only defined up to elements from the
inertia group $T$, and then constructs a representation of $Z/T$,
the factor group of decomposition modulo inertia group; then he
uses this "piece" of the representation for defining the local
factors at ramified primes. Parts of the necessary arguments can
be found in Artin's article on the group theoretic structure of the
discriminant in algebraic number fields that appeared in print in 1931.
In his letter to Hasse from Sept. 18, 1930, Artin gives the following
explanation (the notation is essentially the same as in his articles):
Let ${\mathfrak p}$ be a prime ideal, $\sigma$ the associated
substitution in $K/k$, which is not uniquely determined,
${\mathfrak T}$ the inertia group, and $e$ its order. Set
$$ \chi({\mathfrak p}^\nu) =
\frac{1}{e} \sum_{\tau \in {\mathfrak T}} \chi(\sigma^\nu\tau)\ , $$
which is the mean of all possible values. Then
$$ \log L(s,\chi) =
\sum_{{\mathfrak p},\nu}
\frac{\chi({\mathfrak p}^\nu)}{\nu N{\mathfrak p}^{\nu s}} $$
is the complete definition also for divisors of the discriminant.
$L(s,\chi)$ can be written as usual as a product of the form
$$ L(x,\chi) = \prod_{\mathfrak p}
\frac{1}{|E-N{\mathfrak p}^{-s} A_{\mathfrak p}|}, $$
where $A_{\mathfrak p}$ is a certain matrix attached to
${\mathfrak p}$ (which may be $0$) and only has roots of units
as characteristic roots.
This explains the naive idea behind the definition: since the
Frobenius is not well defined, take the mean over all possible
values. Finally, Noah Snyder has written a very nice
thesis
on Artin L-functions, which contains a translation of Artin's 1923
article on L-series.
Best Answer
One proves explicit formulae essentially by integrating the logarithmic derivative of the $L$-function. For simplicity, let $\psi$ be a Schwartz function with $\psi(1) = 1$, and let $$ \widehat{\psi}(s) = \int_0^\infty \psi(t) t^{s} \frac{dt}{t} $$ be its Mellin transform. Then one can get an explicit formula by considering $$ \sum_{n \geq 1} \Lambda_V(n) \psi(n) = \frac{-1}{2\pi i} \int_{(2)} \frac{L'}{L}(V, s) \widehat{\psi}(s) ds. $$ Typically one would shift the line of integration to $\mathrm{Re}(s) = c < 0$, apply the functional equation, and get an expression of the form $$\begin{align} \sum_{n \geq 1} &\big( \Lambda_V(n) \psi(n) + \Lambda_\overline{V}(n) \tfrac{\psi(n^{-1})}{n} \big) = \\ &\{ \mathrm{polar\;data} \} - \sum_\rho \widehat{\psi}(\rho) + \frac{1}{2\pi i} \int_{(1/2)}\big( \tfrac{\gamma'}{\gamma}(V, s) + \tfrac{\gamma'}{\gamma}(V, 1-s) \big) \widehat{\psi}(s) ds + O(1). \end{align}$$ Here, I take $\overline{V}$ to be the conjugate representation and assume the functional equation is of the form $$ Q^{s/2} \gamma(V, s) L(V, s) = \varepsilon(V) Q^{(1-s)/2} \gamma(V, 1-s) L(\overline{V}, 1-s) $$ for collected gamma factors $\gamma(V, s)$ and a root number $\lvert \varepsilon(V) \rvert = 1$. (Brauer's On Artin's L-series with general group characters details the general functional equations, and this explicit formula is a in $\S$5.5 of Iwaniec–Kowalski).
But a major challenge in the face of being computationally useful is that we don't know the complete polar data for an arbitrary Artin $L$-function. Assuming Artin's conjecture and taking a nontrivial irreducible representation, there should be no polar contribution and we get a classical explicit formula.