Inmathematics, aDirichlet${\displaystyle L}$-series is a function of the form

${\displaystyle L(s,\chi )=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi (n)}{n^s}.}$

where${\displaystyle \chi }$ is aDirichlet character and${\displaystyle s}$ acomplex variable withreal part greater than${\displaystyle 1}$. It is a special case of aDirichlet series. Byanalytic continuation, it can be extended to ameromorphic function on the wholecomplex plane, and is then called aDirichlet${\displaystyle L}$-function and also denoted${\displaystyle L(s,\chi )}$.

These functions are named afterPeter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove thetheorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that${\displaystyle L(s,\chi )}$ is non-zero at${\displaystyle s=1}$. Moreover, if${\displaystyle \chi }$ is principal, then the corresponding Dirichlet${\displaystyle L}$-function has asimple pole at${\displaystyle s=1}$. Otherwise, the${\displaystyle L}$-function isentire.

Since a Dirichlet character${\displaystyle \chi }$  iscompletely multiplicative, its${\displaystyle L}$ -function can also be written as anEuler product in thehalf-plane ofabsolute convergence:

${\displaystyle L(s,\chi )=\prod _p{\left(1-\chi (p)p^{-s}\right)}^{-1}\text{ for }\text{Re}(s)>1,}$ 

where the product is over allprime numbers.^[1]

Results aboutL-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.^[2] This is because of the relationship between a imprimitive character${\displaystyle \chi }$  and the primitive character${\displaystyle {\chi }^{\star }}$  which induces it:^[3]

 

(Here,q is the modulus ofχ.) An application of the Euler product gives a simple relationship between the correspondingL-functions:^[4][5]

${\displaystyle L(s,\chi )=L(s,{\chi }^{\star })\prod _{p|q}\left(1-\frac{{\chi }^{\star }(p)}{p^s}\right)}$ 

(This formula holds for alls, by analytic continuation, even though the Euler product is only valid when Re(s) > 1.) The formula shows that theL-function ofχ is equal to theL-function of the primitive character which inducesχ, multiplied by only a finite number of factors.^[6]

As a special case, theL-function of the principal character${\displaystyle {\chi }_0}$  moduloq can be expressed in terms of theRiemann zeta function:^[7][8]

${\displaystyle L(s,{\chi }_0)=\zeta (s)\prod _{p|q}(1-p^{-s})}$ 

DirichletL-functions satisfy afunctional equation, which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of${\displaystyle L(s,\chi )}$  to the value of${\displaystyle L(1-s,\overline{\chi })}$ . Letχ be a primitive character moduloq, whereq > 1. One way to express the functional equation is:^[9]

${\displaystyle L(s,\chi )=W(\chi )2^s{\pi }^{s-1}{q}^{1/2-s}\sin \left(\frac{\pi }{2}(s+\delta )\right)\mathrm{\Gamma }(1-s)L(1-s,\overline{\chi }).}$ 

In this equation, Γ denotes thegamma function;

${\displaystyle \chi (-1)=(-1{)}^{\delta }}$  ; and

${\displaystyle W(\chi )=\frac{\tau (\chi )}{i^{\delta }\sqrt{q}}}$ 

whereτ ( χ) is aGauss sum:

${\displaystyle \tau (\chi )=\sum _{a=1}^q\chi (a)\exp (2\pi ia/q).}$ 

It is a property of Gauss sums that |τ ( χ) | =q^1/2, so |W ( χ) | = 1.^[10][11]

Another way to state the functional equation is in terms of

${\displaystyle \mathrm{\Lambda }(s,\chi )=q^{s/2}{\pi }^{-(s+\delta )/2}\mathrm{\Gamma }\left(\frac{s+\delta }{2}\right)L(s,\chi ).}$ 

The functional equation can be expressed as:^[9][11]

${\displaystyle \mathrm{\Lambda }(s,\chi )=W(\chi )\mathrm{\Lambda }(1-s,\overline{\chi }).}$ 

The functional equation implies that${\displaystyle L(s,\chi )}$  (and${\displaystyle \mathrm{\Lambda }(s,\chi )}$ ) areentire functions ofs. (Again, this assumes thatχ is primitive character moduloq withq > 1. Ifq = 1, then${\displaystyle L(s,\chi )=\zeta (s)}$  has a pole ats = 1.)^[9][11]

For generalizations, see:Functional equation (L-function).

Letχ be a primitive character moduloq, withq > 1.

There are nozeros ofL(s,χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negativeintegerss:

• Ifχ(−1) = 1, the only zeros ofL(s,χ) with Re(s) < 0 are simple zeros at −2, −4, −6, .... (There is also a zero ats = 0.) These correspond to the poles of${\displaystyle \mathrm{\Gamma }(\frac{s}{2})}$ .^[12]
• Ifχ(−1) = −1, then the only zeros ofL(s,χ) with Re(s) < 0 are simple zeros at −1, −3, −5, .... These correspond to the poles of${\displaystyle \mathrm{\Gamma }(\frac{s+1}{2})}$ .^[12]

These are called the trivial zeros.^[9]

The remaining zeros lie in the critical strip 0 ≤ Re(s) ≤ 1, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line Re(s) = 1/2. That is, if${\displaystyle L(\rho ,\chi )=0}$  then${\displaystyle L(1-\overline{\rho },\chi )=0}$  too, because of the functional equation. Ifχ is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not ifχ is a complex character. Thegeneralized 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 aSiegel 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 DirichletL-functions: for example, forχ a non-real character of modulusq, we have

 

for β + iγ a non-real zero.^[13]

The DirichletL-functions may be written as a linear combination of theHurwitz zeta function at rational values. Fixing an integerk ≥ 1, the DirichletL-functions for characters modulok are linear combinations, with constant coefficients, of theζ(s,a) wherea =r/k andr = 1, 2, ...,k. This means that the Hurwitz zeta function for rationala has analytic properties that are closely related to the DirichletL-functions. Specifically, letχ be a character modulok. Then we can write its DirichletL-function as:^[14]

${\displaystyle L(s,\chi )=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi (n)}{n^s}=\frac{1}{k^s}\sum _{r=1}^k\chi (r)\zeta \left(s,\frac{r}{k}\right).}$ 

• Generalized Riemann hypothesis
• L-function
• Modularity theorem
• Artin conjecture
• Special values of L-functions

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