# Properties

 Label 2-151-151.150-c0-0-2 Degree $2$ Conductor $151$ Sign $1$ Analytic cond. $0.0753588$ Root an. cond. $0.274515$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.24·2-s + 0.554·4-s − 1.80·5-s − 0.554·8-s + 9-s − 2.24·10-s − 0.445·11-s − 1.24·16-s − 0.445·17-s + 1.24·18-s + 1.24·19-s − 0.999·20-s − 0.554·22-s + 2.24·25-s + 1.24·29-s − 1.80·31-s − 0.999·32-s − 0.554·34-s + 0.554·36-s − 0.445·37-s + 1.55·38-s + 1.00·40-s − 1.80·43-s − 0.246·44-s − 1.80·45-s + 1.24·47-s + 49-s + ⋯
 L(s)  = 1 + 1.24·2-s + 0.554·4-s − 1.80·5-s − 0.554·8-s + 9-s − 2.24·10-s − 0.445·11-s − 1.24·16-s − 0.445·17-s + 1.24·18-s + 1.24·19-s − 0.999·20-s − 0.554·22-s + 2.24·25-s + 1.24·29-s − 1.80·31-s − 0.999·32-s − 0.554·34-s + 0.554·36-s − 0.445·37-s + 1.55·38-s + 1.00·40-s − 1.80·43-s − 0.246·44-s − 1.80·45-s + 1.24·47-s + 49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$151$$ Sign: $1$ Analytic conductor: $$0.0753588$$ Root analytic conductor: $$0.274515$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{151} (150, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 151,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8764430973$$ $$L(\frac12)$$ $$\approx$$ $$0.8764430973$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad151 $$1 - T$$
good2 $$1 - 1.24T + T^{2}$$
3 $$1 - T^{2}$$
5 $$1 + 1.80T + T^{2}$$
7 $$1 - T^{2}$$
11 $$1 + 0.445T + T^{2}$$
13 $$1 - T^{2}$$
17 $$1 + 0.445T + T^{2}$$
19 $$1 - 1.24T + T^{2}$$
23 $$1 - T^{2}$$
29 $$1 - 1.24T + T^{2}$$
31 $$1 + 1.80T + T^{2}$$
37 $$1 + 0.445T + T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + 1.80T + T^{2}$$
47 $$1 - 1.24T + T^{2}$$
53 $$1 - T^{2}$$
59 $$1 + 1.80T + T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - 1.24T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$