# Properties

 Label 2-11-1.1-c5-0-1 Degree $2$ Conductor $11$ Sign $1$ Analytic cond. $1.76422$ Root an. cond. $1.32824$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.20·2-s + 16.8·3-s − 27.1·4-s + 75.2·5-s + 37.1·6-s − 225.·7-s − 130.·8-s + 40.5·9-s + 166.·10-s + 121·11-s − 456.·12-s + 455.·13-s − 498.·14-s + 1.26e3·15-s + 579.·16-s + 190.·17-s + 89.5·18-s − 135.·19-s − 2.04e3·20-s − 3.79e3·21-s + 267.·22-s + 2.79e3·23-s − 2.19e3·24-s + 2.53e3·25-s + 1.00e3·26-s − 3.40e3·27-s + 6.11e3·28-s + ⋯
 L(s)  = 1 + 0.390·2-s + 1.08·3-s − 0.847·4-s + 1.34·5-s + 0.421·6-s − 1.73·7-s − 0.721·8-s + 0.166·9-s + 0.525·10-s + 0.301·11-s − 0.915·12-s + 0.747·13-s − 0.679·14-s + 1.45·15-s + 0.565·16-s + 0.160·17-s + 0.0651·18-s − 0.0860·19-s − 1.14·20-s − 1.87·21-s + 0.117·22-s + 1.10·23-s − 0.779·24-s + 0.810·25-s + 0.291·26-s − 0.899·27-s + 1.47·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$11$$ Sign: $1$ Analytic conductor: $$1.76422$$ Root analytic conductor: $$1.32824$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 11,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.645807385$$ $$L(\frac12)$$ $$\approx$$ $$1.645807385$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1 - 121T$$
good2 $$1 - 2.20T + 32T^{2}$$
3 $$1 - 16.8T + 243T^{2}$$
5 $$1 - 75.2T + 3.12e3T^{2}$$
7 $$1 + 225.T + 1.68e4T^{2}$$
13 $$1 - 455.T + 3.71e5T^{2}$$
17 $$1 - 190.T + 1.41e6T^{2}$$
19 $$1 + 135.T + 2.47e6T^{2}$$
23 $$1 - 2.79e3T + 6.43e6T^{2}$$
29 $$1 + 2.60e3T + 2.05e7T^{2}$$
31 $$1 + 1.05e3T + 2.86e7T^{2}$$
37 $$1 - 1.25e4T + 6.93e7T^{2}$$
41 $$1 - 1.13e3T + 1.15e8T^{2}$$
43 $$1 + 1.46e4T + 1.47e8T^{2}$$
47 $$1 + 1.68e4T + 2.29e8T^{2}$$
53 $$1 - 3.31e3T + 4.18e8T^{2}$$
59 $$1 - 1.14e4T + 7.14e8T^{2}$$
61 $$1 + 2.82e4T + 8.44e8T^{2}$$
67 $$1 + 5.14e4T + 1.35e9T^{2}$$
71 $$1 + 1.62e4T + 1.80e9T^{2}$$
73 $$1 + 1.01e4T + 2.07e9T^{2}$$
79 $$1 - 6.08e4T + 3.07e9T^{2}$$
83 $$1 - 4.57e4T + 3.93e9T^{2}$$
89 $$1 + 8.22e4T + 5.58e9T^{2}$$
97 $$1 - 5.30e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$