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Let $\rho: \operatorname{Gal}(K/F) \rightarrow \operatorname{GL}_n(\mathbb C)$ be a representation for an unramified extension $K/F$ of $p$-adic fields. Let $\operatorname{Frob}_{K/F}$ be the (arithmetic) Frobenius element of $K/F$. Artin defined the local L-function $L(s,\rho)$ by

$$L(s,\rho) = \operatorname{det}(I_n - \rho(\operatorname{Frob}_{K/F})q^{-s})^{-1}$$

where $q$ is the number of elements in the residue field of $F$.

Nowadays, it's more popular to define L-functions use the geometric (inverse) Frobenius. That is, a lot of people myself included would rather say that

$$L(s,\rho) = \operatorname{det}(I_n - \rho(\operatorname{Frob}_{K/F})^{-1}q^{-s})^{-1}.$$

Which definition does Langlands use in his papers? I can't tell. Here is where he defines Artin L-functions in his paper On Artin's L-functions:

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He says that for a representation $\sigma$ of dimension one, the Artin L-function $L(s,\sigma)$ is defined to be the Tate L-function $L(s,\omega)$, if $\omega$ is the quasicharacter of $C_F = F^{\ast}$ corresponding to $\sigma$. But what is the correspondence $\sigma \mapsto \omega$? Langlands does not say.

It probably comes from however Langlands is choosing to identify of $F^{\ast}$ with the abelianized Weil group $W_F^{\operatorname{ab}}$ (which I have asked about in a previous question), but I do not know for sure.

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    $\begingroup$ You cannot redefine Artin $L$-functions, just as you cannot redefine Dirichlet $L$-functions and the Riemann zeta function. These objects have been with us for a century or more, so you must respect them and stick to them. I am 99% sure that Langlands would agree with this, and he uses the first display as the definition of Artin $L$-functions. The second display is also legitimate, of course: it is the dual of the Artin $L$-function. My two cents. $\endgroup$
    – GH from MO
    Commented Apr 5, 2020 at 6:42

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