Functional equation (L-function)

A functional equation, in the context of L-functions, is a fundamental relation that connects the values of the L-function at different points in the complex plane. More specifically, it relates the value of the L-function L(s) at a complex number s to its value at k - s, where k is a constant, usually an integer, referred to as the center of symmetry or critical line of the functional equation. The presence of such a functional equation is a defining characteristic of an L-function.

The typical form of a functional equation involves not just L(s) and L(k-s), but also gamma functions, exponential factors, and possibly the complex conjugate of L(s). These additional factors are crucial for ensuring that the equation holds across the entire complex plane (or at least a substantial region thereof) and are related to the analytic properties of the L-function. The functional equation essentially encodes deep arithmetic or geometric properties of the object that the L-function is associated with.

More formally, a functional equation for an L-function L(s) often takes the form:

Λ(s) = ω Λ̄(k-s)

where:

• L(s) is the L-function.
• s is a complex variable.
• k is the center of symmetry (often an integer).
• Λ(s) is the completed L-function, which is the original L-function L(s) multiplied by appropriate gamma factors, exponential factors, and other terms related to the arithmetic conductor of the L-function.
• ω is a complex number of absolute value 1 (i.e., |ω| = 1), called the root number. It represents the sign of the functional equation and often carries arithmetic significance.
• Λ̄(s) denotes the complex conjugate of Λ(s).

The functional equation's main importance lies in its ability to extend the definition of the L-function beyond its initial domain of convergence. L-functions are typically defined by a Dirichlet series that converges only for complex numbers s with sufficiently large real part. The functional equation allows us to analytically continue the L-function to the entire complex plane (or at least a meromorphic continuation) and provides a relationship between values of the L-function on opposite sides of the critical line, Re(s) = k/2.

The functional equation is vital for proving results about the distribution of prime numbers (or, more generally, the arithmetic objects associated with the L-function), such as the Prime Number Theorem. It plays a crucial role in the proof of deep results related to L-functions, such as the Riemann Hypothesis (which conjectures that all non-trivial zeros of the Riemann zeta function lie on the critical line) and related conjectures like the Generalized Riemann Hypothesis.

Different types of L-functions possess different functional equations, with the precise form of the equation depending on the arithmetic or geometric object associated with the L-function. For instance, the Riemann zeta function, Dirichlet L-functions, modular forms, and elliptic curves each have L-functions associated with them, and each of those L-functions satisfies its own functional equation. Understanding the functional equation is essential to understanding the analytic and arithmetic properties of these objects.