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A Dirichlet L-function is given by ${\displaystyle L(s, \chi)=\sum_{n=1}^\infty \frac {\chi(n)}{n^s}}$ for Re$(s) >1$, where $\chi$ is a Dirichlet character.

These L-functions can be meromorphically continued to the complex plane and have an Euler product ${\displaystyle L(s, \chi)= \prod_p (1 - \chi(p) p^{-s} )^{-1}}$, where the product is over all primes $p$.

The functional equation takes the form $\Lambda(s,\chi) = q^{s/2} \Gamma_{\mathbb R} (s+a) L(s,\chi) = \varepsilon_\chi \overline{\Lambda}(1-s)$, where $q$ is the conductor of $\chi$.

These L-functions were introduced by Dirichlet in the mid-1800s as a tool to study prime numbers in arithmetic progressions.

Knowl status:
  • Review status: reviewed
  • Last edited by Stephan Ehlen on 2019-04-30 08:28:15
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