# Properties

 Label 1-1-1.1-r0-0-0 Degree $1$ Conductor $1$ Sign $1$ Analytic cond. $0.00464398$ Root an. cond. $0.00464398$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

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. Its unique pole is located at $s=1$.

## Dirichlet series

 L(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 + ⋯
 L(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}\Lambda(s)=\mathstrut &\Gamma_{\R}(s) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1$$ Sign: $1$ Analytic conductor: $$0.00464398$$ Root analytic conductor: $$0.00464398$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 1,\ (0:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$-1.460354508$$ $$L(\frac12)$$ $$\approx$$ $$-1.460354508$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
good2 $$1 - T$$
3 $$1 - T$$
5 $$1 - T$$
7 $$1 - T$$
11 $$1 - T$$
13 $$1 - T$$
17 $$1 - T$$
19 $$1 - T$$
23 $$1 - T$$
29 $$1 - T$$
31 $$1 - T$$
37 $$1 - T$$
41 $$1 - T$$
43 $$1 - T$$
47 $$1 - T$$
53 $$1 - T$$
59 $$1 - T$$
61 $$1 - T$$
67 $$1 - T$$
71 $$1 - T$$
73 $$1 - T$$
79 $$1 - T$$
83 $$1 - T$$
89 $$1 - T$$
97 $$1 - T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$