Dirichlet L-function - Functional Equation

Functional Equation

Let us assume that χ is a primitive character to the modulus k. Defining

$\Lambda(s,\chi) = \left(\frac{\pi}{k}\right)^{-(s+a)/2} \Gamma\left(\frac{s+a}{2}\right) L(s,\chi),$

where Γ denotes the Gamma function and the symbol a is given by

one has the functional equation

Here we wrote τ(χ) for the Gauss sum

Note that |τ(χ)|=k1/2.

Read more about this topic: Dirichlet L-function

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