Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler product in the half-plane of absolute 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 all prime numbers.^[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.^[2] This is because of the relationship between a imprimitive character ${\displaystyle \chi }$  and the primitive character ${\displaystyle \chi ^{\star }}$  which induces it:^[3]

${\displaystyle \chi (n)={\begin{cases}\chi ^{\star }(n),&\mathrm {if} \gcd(n,q)=1\\0,&\mathrm {if} \gcd(n,q)\neq 1\end{cases}}}$ 

(Here, q is the modulus of χ.) An application of the Euler product gives a simple relationship between the corresponding L-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 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 ${\displaystyle \chi _{0}}$  modulo q can be expressed in terms of the Riemann zeta function:^[7][8]

${\displaystyle L(s,\chi _{0})=\zeta (s)\prod _{p\,|\,q}(1-p^{-s})}$ 

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 ${\displaystyle L(s,\chi )}$  to the value of ${\displaystyle L(1-s,{\overline {\chi }})}$ . Let χ be a primitive character modulo q, where q > 1. One way to express the functional equation is:^[9]

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

In this equation, Γ denotes the gamma function;

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

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

where τ ( χ) is a Gauss 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

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

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

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

The functional equation implies that ${\displaystyle L(s,\chi )}$  (and ${\displaystyle \Lambda (s,\chi )}$ ) are entire functions of s. (Again, this assumes that χ is primitive character modulo q with q > 1. If q = 1, then ${\displaystyle L(s,\chi )=\zeta (s)}$  has a pole at s = 1.)^[9][11]

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

Let χ be a primitive character modulo q, with q > 1.

There are no zeros of L(s, χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negative integers s:

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

${\displaystyle \beta <1-{\frac {c}{\log \!\!\;{\big (}q(2+|\gamma |){\big )}}}\ }$ 

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]

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

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

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