Euler product and determinant local L-factor

Let $F_v$ be a nonarchimedean local field with residue cardinality $q_v$ , let $\pi_v$ be unramified with Satake parameter $c(\pi_v)$ , and let

r:{}^LG\longrightarrow\operatorname{GL}(V_r)

be a finite-dimensional algebraic representation of the $L$ -group. The unramified local $L$ -factor is

L_v(s,\pi_v,r) = \det\!\left( 1-r(c(\pi_v))q_v^{-s}\mid V_r \right)^{-1}.

It depends only on the semisimple conjugacy class and on the chosen Frobenius and normalization conventions.

Euler product

For a global automorphic representation $\pi=\bigotimes_v'\pi_v$ , written as a restricted tensor product, let $S$ contain the archimedean places and all places where the data are ramified. The incomplete Euler product is

L^S(s,\pi,r)=\prod_{v\notin S}L_v(s,\pi_v,r).

It converges absolutely in a right half-plane in the standard automorphic settings. A completed $L$ -function includes specified ramified factors and archimedean gamma factors.

Ramified Weil-Deligne factor

Suppose the composite of a local parameter with $r$ gives the Weil–Deligne representation $(\rho_{v,r},N_{v,r})$ . With arithmetic Frobenius in the displayed formula, one common convention is

L_v(s,\pi_v,r) = \det\!\left( 1-q_v^{-s}\rho_{v,r}(\operatorname{Frob}_v) \;\middle|\; (\ker N_{v,r})^{I_v} \right)^{-1}.

Using geometric Frobenius replaces the operator by its inverse. Different normalizations of local Langlands can also insert a norm twist.

What the letter could not yet encode

The letter's unramified determinant formula is foundational, but a canonical ramified factor needs the later Weil–Deligne and local Langlands formalism. Equality of almost-all unramified factors is weaker than full local compatibility because it does not recover the inertia action or monodromy.

References

Pierre Deligne, “Les constantes des équations fonctionnelles des fonctions $L$ ,” in Modular Functions of One Variable II, 1973.
A. Borel, “Automorphic $L$ -functions,” Proc. Sympos. Pure Math. 33, part 2, 1979.