# Properties

 Degree 1 Conductor $1$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes

# Related objects

The Riemann zeta function is the prototypical L-function. It is the only L-function of degree 1 and conductor 1, and (conjecturally) it is the only primitive L-function with a pole.

## Dirichlet series

 $\zeta(s)$  = 1 + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯

## Functional equation

\begin{aligned}\xi(s)=\mathstrut &\Gamma_{\R}(s) \, \zeta(s)\cr=\mathstrut & \,\xi(1-s)\end{aligned}

## Invariants

 $$d$$ = $$1$$ $$N$$ = $$1$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $$(1,\ 1,\ (0:\ ),\ 1)$$

## Euler product

\begin{aligned}\zeta(s) = \prod_p (1 - p^{-s})^{-1}\end{aligned}

## Particular Values

$\zeta(1/2) \approx -1.460354508$
Pole at $$s=1$$