I'm trying to figure out what the Artin Conductor of a representation is using Chapter IV.$2\text-3$ of Serre's Local Fields, and I'm struggling to understand the motivation behind its definition.
If $L/K$ is a finite Galois extension of local fields with integer rings $\mathcal O_L=\mathcal O_K[x]$, $G = \mathrm{Gal}(L/K)$ and $f = [\overline L:\overline K]$ is the residue degree, then we define $i_G(s) = v_K(s(x)-x)$ for $s\in G$ and $$a_G(s) =\begin{cases}-f\cdot i_G(s) & \text{if }s \ne 1\\\\f\displaystyle\sum_{s\ne 1}i_G(s)&\text{if }s=1\end{cases}$$
We then define for any class function $\varphi$ $$f(\varphi) = (\varphi,a_G)$$ so that $$a_G = \sum_{\chi}f(\chi)\chi$$ as a sum of irreducible characters. We then deduce some key properties about $f$ - in particular that we have
$$f(\chi)=(\chi,a_G)=\sum_i\frac1{(G_0:G_i)}\dim(V/V^{G_i})$$ where $G_i$ is the $i^{\text{th}}$ ramification group of $G$ and $\chi$ is the character of a representation $\rho:G\to \mathrm{GL}(V)$. We can then define the local conductor of $\chi$ to be the ideal $\mathfrak f(\chi)=\mathfrak p_K^{f(\chi)}$ which we can extend to the global conductor by taking a product over primes.
My question is:
What motivates this definition?
Once we have defined $a_G$, the definition of $f$ seems natural; I'm unclear as to why we would define $a_G$ in the first place, and why we would then use it to construct an ideal $\mathfrak f(\chi)$.
One important motivation is the conductor-discriminant formula, which says that for $L/K$ a finite Galois extension of number fields, we have that the discriminant of $L$ over $K$ is equal to the product $\mathfrak f(\chi)^{\chi(1)},$ where $\chi$ runs over all irred. characters of the Galois group of $L/K$.
Maybe more relevant to you is the local version of this formula, which takes exactly the same form, except that now $L/K$ should be a finite Galois extension of non-archimedean local fields.