# Properties

 Label 2-231-1.1-c3-0-17 Degree $2$ Conductor $231$ Sign $-1$ Analytic cond. $13.6294$ Root an. cond. $3.69180$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.561·2-s − 3·3-s − 7.68·4-s + 6.68·5-s + 1.68·6-s + 7·7-s + 8.80·8-s + 9·9-s − 3.75·10-s − 11·11-s + 23.0·12-s + 14.3·13-s − 3.93·14-s − 20.0·15-s + 56.5·16-s − 47.7·17-s − 5.05·18-s + 11.9·19-s − 51.3·20-s − 21·21-s + 6.17·22-s − 44.4·23-s − 26.4·24-s − 80.3·25-s − 8.03·26-s − 27·27-s − 53.7·28-s + ⋯
 L(s)  = 1 − 0.198·2-s − 0.577·3-s − 0.960·4-s + 0.597·5-s + 0.114·6-s + 0.377·7-s + 0.389·8-s + 0.333·9-s − 0.118·10-s − 0.301·11-s + 0.554·12-s + 0.305·13-s − 0.0750·14-s − 0.345·15-s + 0.883·16-s − 0.681·17-s − 0.0661·18-s + 0.143·19-s − 0.574·20-s − 0.218·21-s + 0.0598·22-s − 0.403·23-s − 0.224·24-s − 0.642·25-s − 0.0605·26-s − 0.192·27-s − 0.363·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$231$$    =    $$3 \cdot 7 \cdot 11$$ Sign: $-1$ Analytic conductor: $$13.6294$$ Root analytic conductor: $$3.69180$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 231,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 3T$$
7 $$1 - 7T$$
11 $$1 + 11T$$
good2 $$1 + 0.561T + 8T^{2}$$
5 $$1 - 6.68T + 125T^{2}$$
13 $$1 - 14.3T + 2.19e3T^{2}$$
17 $$1 + 47.7T + 4.91e3T^{2}$$
19 $$1 - 11.9T + 6.85e3T^{2}$$
23 $$1 + 44.4T + 1.21e4T^{2}$$
29 $$1 + 139.T + 2.43e4T^{2}$$
31 $$1 + 208.T + 2.97e4T^{2}$$
37 $$1 + 253.T + 5.06e4T^{2}$$
41 $$1 + 156.T + 6.89e4T^{2}$$
43 $$1 + 263.T + 7.95e4T^{2}$$
47 $$1 - 386.T + 1.03e5T^{2}$$
53 $$1 + 36.5T + 1.48e5T^{2}$$
59 $$1 + 114.T + 2.05e5T^{2}$$
61 $$1 + 53.0T + 2.26e5T^{2}$$
67 $$1 - 132.T + 3.00e5T^{2}$$
71 $$1 - 583.T + 3.57e5T^{2}$$
73 $$1 + 817.T + 3.89e5T^{2}$$
79 $$1 + 369.T + 4.93e5T^{2}$$
83 $$1 + 69.1T + 5.71e5T^{2}$$
89 $$1 - 467.T + 7.04e5T^{2}$$
97 $$1 + 1.17e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$