Root number - Knowlpedia

Definition

After an $L$ -function has been completed and normalized so that its functional equation relates $s$ to $1-s$ , its global root number is the unit-modulus constant $w$ in

\Lambda(s)=w\,\Lambda(1-s,\text{dual data}).

Here “dual data” means the appropriate contragredient parameter or representation. For self-dual data with the usual reality conditions, $w\in\{+1,-1\}$ and is often called the sign of the functional equation. Without self-duality the root number can be any complex number of absolute value one.

Local root numbers

For a local Weil–Deligne representation $V$ and additive character $\psi$ , the local root number is the unit-modulus normalization of the epsilon factor at the central point,

w(V,\psi)= \frac{\varepsilon(1/2,V,\psi)} {|\varepsilon(1/2,V,\psi)|}.

For global data and compatible choices, the global root number is the product of the local root numbers. Almost every local factor equals $1$ .

Convention warning

The central point may be written $1/2$ , $0$ , or another shifted value, depending on whether the L-function is in analytic, motivic, or unitary normalization. Local root numbers can also change with the additive character; the global product is choice-independent after the standard global compatibility is imposed.

References

Pierre Deligne, “Les constantes des équations fonctionnelles des fonctions $L$ ,” in Modular Functions of One Variable II, Lecture Notes in Mathematics 349, Springer, 1973, 501–597.
John Tate, “Number theoretic background,” in Automorphic Forms, Representations and L-Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979.