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

Famous quotes containing the words functional and/or equation:

“In short, the building becomes a theatrical demonstration of its functional ideal. In this romanticism, High-Tech architecture is, of course, no different in spirit—if totally different in form—from all the romantic architecture of the past.”
—Dan Cruickshank (b. 1949)

“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)