# Properties

 Label 2-143-143.142-c0-0-0 Degree $2$ Conductor $143$ Sign $1$ Analytic cond. $0.0713662$ Root an. cond. $0.267144$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.61·2-s − 1.61·3-s + 1.61·4-s + 2.61·6-s + 0.618·7-s − 8-s + 1.61·9-s + 11-s − 2.61·12-s + 13-s − 1.00·14-s − 2.61·18-s − 1.61·19-s − 1.00·21-s − 1.61·22-s + 0.618·23-s + 1.61·24-s + 25-s − 1.61·26-s − 27-s + 1.00·28-s + 32-s − 1.61·33-s + 2.61·36-s + 2.61·38-s − 1.61·39-s + 0.618·41-s + ⋯
 L(s)  = 1 − 1.61·2-s − 1.61·3-s + 1.61·4-s + 2.61·6-s + 0.618·7-s − 8-s + 1.61·9-s + 11-s − 2.61·12-s + 13-s − 1.00·14-s − 2.61·18-s − 1.61·19-s − 1.00·21-s − 1.61·22-s + 0.618·23-s + 1.61·24-s + 25-s − 1.61·26-s − 27-s + 1.00·28-s + 32-s − 1.61·33-s + 2.61·36-s + 2.61·38-s − 1.61·39-s + 0.618·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$143$$    =    $$11 \cdot 13$$ Sign: $1$ Analytic conductor: $$0.0713662$$ Root analytic conductor: $$0.267144$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{143} (142, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 143,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.2259201697$$ $$L(\frac12)$$ $$\approx$$ $$0.2259201697$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1 - T$$
13 $$1 - T$$
good2 $$1 + 1.61T + T^{2}$$
3 $$1 + 1.61T + T^{2}$$
5 $$1 - T^{2}$$
7 $$1 - 0.618T + T^{2}$$
17 $$1 - T^{2}$$
19 $$1 + 1.61T + T^{2}$$
23 $$1 - 0.618T + T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 - 0.618T + T^{2}$$
43 $$1 - T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + 1.61T + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + 1.61T + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - 0.618T + T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$