# Properties

 Label 1-5269-5269.5268-r0-0-0 Degree $1$ Conductor $5269$ Sign $1$ Analytic cond. $24.4691$ Root an. cond. $24.4691$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + 29-s − 30-s + ⋯
 L(s)  = 1 − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + 29-s − 30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$5269$$    =    $$11 \cdot 479$$ Sign: $1$ Analytic conductor: $$24.4691$$ Root analytic conductor: $$24.4691$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{5269} (5268, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 5269,\ (0:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.441272559$$ $$L(\frac12)$$ $$\approx$$ $$2.441272559$$ $$L(1)$$ $$\approx$$ $$1.293706842$$ $$L(1)$$ $$\approx$$ $$1.293706842$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1$$
479 $$1$$
good2 $$1 - T$$
3 $$1 + T$$
5 $$1 + T$$
7 $$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}$$