Dirichlet L-function

a complex variable with real part greater than

It is a special case of a Dirichlet series.

By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove the theorem on primes in arithmetic progressions that also bears his name.

[1] Results about L-functions are often stated more simply if the character is assumed to be primitive, although the results typically can be extended to imprimitive characters with minor complications.

An application of the Euler product gives a simple relationship between the corresponding L-functions:[4][5] (This formula holds for all s, by analytic continuation, even though the Euler product is only valid when Re(s) > 1.)

The formula shows that the L-function of χ is equal to the L-function of the primitive character which induces χ, multiplied by only a finite number of factors.

[6] As a special case, the L-function of the principal character

modulo q can be expressed in terms of the Riemann zeta function:[7][8] Dirichlet L-functions satisfy a functional equation, which provides a way to analytically continue them throughout the complex plane.

The functional equation relates the value of

One way to express the functional equation is:[9] In this equation, Γ denotes the gamma function; where τ ( χ) is a Gauss sum: It is a property of Gauss sums that |τ ( χ) | = q1/2, so |W ( χ) | = 1.

) are entire functions of s. (Again, this assumes that χ is primitive character modulo q with q > 1.

)[9][11] For generalizations, see: Functional equation (L-function).

The non-trivial zeros are symmetrical about the critical line Re(s) = 1/2.

If χ is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if χ is a complex character.

The generalized Riemann hypothesis is the conjecture that all the non-trivial zeros lie on the critical line Re(s) = 1/2.

[9] Up to the possible existence of a Siegel zero, zero-free regions including and beyond the line Re(s) = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet L-functions: for example, for χ a non-real character of modulus q, we have for β + iγ a non-real zero.

[13] The Dirichlet L-functions may be written as a linear combination of the Hurwitz zeta function at rational values.

Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ(s,a) where a = r/k and r = 1, 2, ..., k. This means that the Hurwitz zeta function for rational a has analytic properties that are closely related to the Dirichlet L-functions.

Specifically, let χ be a character modulo k. Then we can write its Dirichlet L-function as:[14]