# Properties

 Degree $2$ Conductor $2169$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.35·2-s − 0.160·4-s − 2.74·5-s − 0.283·7-s − 2.93·8-s − 3.71·10-s + 4.12·11-s + 0.0271·13-s − 0.384·14-s − 3.65·16-s + 1.28·17-s − 5.72·19-s + 0.440·20-s + 5.59·22-s + 5.97·23-s + 2.51·25-s + 0.0368·26-s + 0.0455·28-s + 2.55·29-s − 2.02·31-s + 0.905·32-s + 1.74·34-s + 0.777·35-s + 2.42·37-s − 7.76·38-s + 8.03·40-s + 11.0·41-s + ⋯
 L(s)  = 1 + 0.959·2-s − 0.0802·4-s − 1.22·5-s − 0.107·7-s − 1.03·8-s − 1.17·10-s + 1.24·11-s + 0.00754·13-s − 0.102·14-s − 0.913·16-s + 0.312·17-s − 1.31·19-s + 0.0984·20-s + 1.19·22-s + 1.24·23-s + 0.503·25-s + 0.00723·26-s + 0.00860·28-s + 0.474·29-s − 0.364·31-s + 0.160·32-s + 0.299·34-s + 0.131·35-s + 0.399·37-s − 1.26·38-s + 1.27·40-s + 1.72·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2169$$    =    $$3^{2} \cdot 241$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{2169} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2169,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.850353682$$ $$L(\frac12)$$ $$\approx$$ $$1.850353682$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
241 $$1 + T$$
good2 $$1 - 1.35T + 2T^{2}$$
5 $$1 + 2.74T + 5T^{2}$$
7 $$1 + 0.283T + 7T^{2}$$
11 $$1 - 4.12T + 11T^{2}$$
13 $$1 - 0.0271T + 13T^{2}$$
17 $$1 - 1.28T + 17T^{2}$$
19 $$1 + 5.72T + 19T^{2}$$
23 $$1 - 5.97T + 23T^{2}$$
29 $$1 - 2.55T + 29T^{2}$$
31 $$1 + 2.02T + 31T^{2}$$
37 $$1 - 2.42T + 37T^{2}$$
41 $$1 - 11.0T + 41T^{2}$$
43 $$1 - 10.4T + 43T^{2}$$
47 $$1 + 4.54T + 47T^{2}$$
53 $$1 - 9.30T + 53T^{2}$$
59 $$1 - 9.94T + 59T^{2}$$
61 $$1 - 8.17T + 61T^{2}$$
67 $$1 - 4.40T + 67T^{2}$$
71 $$1 - 3.80T + 71T^{2}$$
73 $$1 + 15.6T + 73T^{2}$$
79 $$1 - 6.69T + 79T^{2}$$
83 $$1 - 4.32T + 83T^{2}$$
89 $$1 + 0.746T + 89T^{2}$$
97 $$1 - 11.9T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$