# Properties

 Label 1-5-5.4-r0-0-0 Degree $1$ Conductor $5$ Sign $1$ Analytic cond. $0.0232199$ Root an. cond. $0.0232199$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

After the Riemann zeta function, the analytic conductor of this L-function is the smallest among L-functions of degree 1.

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 22-s − 23-s + 24-s + 26-s − 27-s − 28-s + 29-s + 31-s − 32-s − 33-s + 34-s + ⋯
 L(s)  = 1 − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 22-s − 23-s + 24-s + 26-s − 27-s − 28-s + 29-s + 31-s − 32-s − 33-s + 34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$5$$ Sign: $1$ Analytic conductor: $$0.0232199$$ Root analytic conductor: $$0.0232199$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{5} (4, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 5,\ (0:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.2317509475$$ $$L(\frac12)$$ $$\approx$$ $$0.2317509475$$ $$L(1)$$ $$\approx$$ $$0.4304089409$$ $$L(1)$$ $$\approx$$ $$0.4304089409$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
good2 $$1 - T$$
3 $$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}$$