# Properties

 Label 2-231-1.1-c1-0-6 Degree $2$ Conductor $231$ Sign $1$ Analytic cond. $1.84454$ Root an. cond. $1.35814$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.79·2-s − 3-s + 1.20·4-s + 3·5-s − 1.79·6-s + 7-s − 1.41·8-s + 9-s + 5.37·10-s − 11-s − 1.20·12-s + 13-s + 1.79·14-s − 3·15-s − 4.95·16-s + 7.58·17-s + 1.79·18-s − 6.58·19-s + 3.62·20-s − 21-s − 1.79·22-s − 5.58·23-s + 1.41·24-s + 4·25-s + 1.79·26-s − 27-s + 1.20·28-s + ⋯
 L(s)  = 1 + 1.26·2-s − 0.577·3-s + 0.604·4-s + 1.34·5-s − 0.731·6-s + 0.377·7-s − 0.501·8-s + 0.333·9-s + 1.69·10-s − 0.301·11-s − 0.348·12-s + 0.277·13-s + 0.478·14-s − 0.774·15-s − 1.23·16-s + 1.83·17-s + 0.422·18-s − 1.51·19-s + 0.810·20-s − 0.218·21-s − 0.381·22-s − 1.16·23-s + 0.289·24-s + 0.800·25-s + 0.351·26-s − 0.192·27-s + 0.228·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$231$$    =    $$3 \cdot 7 \cdot 11$$ Sign: $1$ Analytic conductor: $$1.84454$$ Root analytic conductor: $$1.35814$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 231,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.153673043$$ $$L(\frac12)$$ $$\approx$$ $$2.153673043$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + T$$
7 $$1 - T$$
11 $$1 + T$$
good2 $$1 - 1.79T + 2T^{2}$$
5 $$1 - 3T + 5T^{2}$$
13 $$1 - T + 13T^{2}$$
17 $$1 - 7.58T + 17T^{2}$$
19 $$1 + 6.58T + 19T^{2}$$
23 $$1 + 5.58T + 23T^{2}$$
29 $$1 + 8.16T + 29T^{2}$$
31 $$1 - 3.58T + 31T^{2}$$
37 $$1 - T + 37T^{2}$$
41 $$1 + 11.1T + 41T^{2}$$
43 $$1 - 1.58T + 43T^{2}$$
47 $$1 - 1.41T + 47T^{2}$$
53 $$1 + 9.58T + 53T^{2}$$
59 $$1 - 4.58T + 59T^{2}$$
61 $$1 - 10T + 61T^{2}$$
67 $$1 - 8.58T + 67T^{2}$$
71 $$1 - 11.1T + 71T^{2}$$
73 $$1 - 7T + 73T^{2}$$
79 $$1 - 7.16T + 79T^{2}$$
83 $$1 + 11.5T + 83T^{2}$$
89 $$1 - 9.16T + 89T^{2}$$
97 $$1 + 2.41T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$