Let \(\rho:\mathrm{Gal}(K/F)\to \mathrm{GL}(V)\) be an Artin representation, the Artin L-function is
\[L(\rho,s) = \sum_{n=1}^\infty \frac{a_n}{n^s} = \prod_{p\text{ prime of }F} \frac{1}{F_p(Np^{-s})}\]
where \(F_p(T) = \det(1-\rho(\mathrm{Frob}_p)T|V^{I_p})\), and \(Np = [\mathcal{O}_F:\mathbb{F}_p]\).
here \(V^{I_p}\) are inertia invariants. For unramified \(p\) it is just \(V\).
unfortunately these \(a_n\) does not have a very well-known explanation, so we are unable to use Poisson summation to get a functional equation.
Exercise
This is independent of the choice of \(I_p\) and \(\mathrm{Frob}_p\).
Writing \[L(\rho,s) = \prod_{p\text{ prime of }\mathbb{Q}} \prod_{\mathfrak{p}|p} \frac{1}{F_\mathfrak{p}(N\mathfrak{p}^{-s})}\], check that \(L(\rho,s)\) is a \(L\)-function of degree \(\dim V\cdot [F:\mathbb{Q}]\).
Example
\[\Delta:\mathrm{Gal}\left(\mathbb{Q}\left(\zeta_3,2^{\frac{1}{3}}\right)/\mathbb{Q}\right)\to \mathrm{GL}_2(V), \quad \dim V = 2\]
We can find a representation of \(S_3\) that maps
\[1\mapsto \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}, \quad (23)\mapsto \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}, \quad (123)\mapsto \begin{pmatrix} \zeta_3 & 0\\ 0 & \zeta_3^2 \end{pmatrix}\]
The corresponding factors are \((1-T)^2, 1-T^2, 1+T+T^2\).
\[L = 1\cdot 1\cdot \frac{1}{(1-5^{-s})(1+5^{-s})} \cdot \frac{1}{1+7^{-s}+7^{-2s}}\cdots\]
Exercise
Find some \(\Delta\) and \(L(\Delta,s)\) on LMFDB.
Theorem. There is a bijection
\[\{\text{primitive Dirichlet characters}\} \leftrightarrow \{1\text{-d Artin representations of }\mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})\}\]
Such that
\(\chi\text{ modulus }m\Leftrightarrow \rho_\chi\) factoring through \(\mathrm{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q})\) and not for smaller \(n|m\).
\(L(\chi,s)=L(\rho_\chi,s)\).
Proof.
Take a character \(\chi:\mathbb{Z}_m^\times\to \mathbb{C}^\times\) that is primitive, let
\[\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q})\cong (\mathbb{Z}/m\mathbb{Z})^\times \quad (\zeta_m\mapsto \zeta_m^a)\mapsto a^{-1}\]
be the canonical isomorphism, compose it with \(\chi:(\mathbb{Z}/m\mathbb{Z})^\times\to \mathbb{C}^\times\) we get a representation \(\mathrm{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q})\to \mathbb{C}^\times\).
Conversely if \(\rho:G_\mathbb{Q}\to \mathbb{C}^\times\) is a \(1\)-dimensional Artin representation, let \(N=\ker\rho\) be normal of finite index and \(K=\overline{\mathbb{Q}}^N\) finite Galois, we have
\[\mathrm{Gal}(K/\mathbb{Q})\cong \mathrm{Im}\rho\subset \mathbb{C}^\times\]
is Abelian, so \(K\) is contained in some cyclotomic field \(\mathbb{Q}(\zeta_m)\), and \(\rho\) factors through \(\mathrm{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q})\) for some \(m\), therefore \(\rho\cong \rho_\chi\) for some \(\chi\).
Let’s compare \(L(\chi,s)\) and \(L(\rho_\chi,s)\) by comparing the local factors.
Take \(\chi\) of modulus \(m\) primitive, \(p\nmid m\Leftrightarrow p\) unramified in \(\mathbb{Q}(\zeta_m)/\mathbb{Q}\Leftrightarrow I_p=1\).
On the left we have \(F_p(T)_L = 1-\chi(p)T\).
On the right, \[F_p(T)_R = \det(1-\rho_\chi(\mathrm{Frob}_p^{-1})T | V_{\dim = 1}^{I_p}) = 1-\rho_\chi(\mathrm{Frob}_p^{-1})T\]
Interestingly, since our Frobenius is act by raising to the power of \(p\), \(\mathrm{Frob}_p^{-1}\) acts as \(\zeta\mapsto \zeta^{p^{-1}}\), under our correspondence corresponds to \(p\in (\mathbb{Z}/m\mathbb{Z})^\times\).
The same holds for any number field \(F\), the Hecke characters over \(F\) of trivial \(\infty\) type corresponds to \(1\)-dimensional Artin representations of \(\mathrm{Gal}(\overline{F}/F)\). The proof of this goes to CFT.
Let \(G\) be a finite group, \(X\) a \(G\)-set, we have a representation \(\mathbb{C}[X]\)
Note \(\mathbb{C}[X]^G\cong \mathbb{C}^{\#{\text{orbits of }G\text{ on }X}}\)
Theorem. Let \(M/F\) be finite, \(K\) Galois closure.
\[X=\{F-\text{embedding} M\hookrightarrow K\}\]
\(V=\mathbb{C}[X]\) associated permutation representation of \(G\)