# Properties

 Label 2-11-11.10-c8-0-2 Degree $2$ Conductor $11$ Sign $1$ Analytic cond. $4.48116$ Root an. cond. $2.11687$ Motivic weight $8$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 113·3-s + 256·4-s + 1.15e3·5-s + 6.20e3·9-s + 1.46e4·11-s − 2.89e4·12-s − 1.30e5·15-s + 6.55e4·16-s + 2.94e5·20-s − 5.31e5·23-s + 9.34e5·25-s + 3.98e4·27-s − 1.54e6·31-s − 1.65e6·33-s + 1.58e6·36-s + 7.16e5·37-s + 3.74e6·44-s + 7.14e6·45-s − 6.08e6·47-s − 7.40e6·48-s + 5.76e6·49-s − 1.52e7·53-s + 1.68e7·55-s − 4.10e6·59-s − 3.32e7·60-s + 1.67e7·64-s + 1.98e7·67-s + ⋯
 L(s)  = 1 − 1.39·3-s + 4-s + 1.84·5-s + 0.946·9-s + 11-s − 1.39·12-s − 2.56·15-s + 16-s + 1.84·20-s − 1.90·23-s + 2.39·25-s + 0.0750·27-s − 1.66·31-s − 1.39·33-s + 0.946·36-s + 0.382·37-s + 44-s + 1.74·45-s − 1.24·47-s − 1.39·48-s + 49-s − 1.93·53-s + 1.84·55-s − 0.338·59-s − 2.56·60-s + 64-s + 0.982·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$11$$ Sign: $1$ Analytic conductor: $$4.48116$$ Root analytic conductor: $$2.11687$$ Motivic weight: $$8$$ Rational: yes Arithmetic: yes Character: $\chi_{11} (10, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 11,\ (\ :4),\ 1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.573596594$$ $$L(\frac12)$$ $$\approx$$ $$1.573596594$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1 - p^{4} T$$
good2 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
3 $$1 + 113 T + p^{8} T^{2}$$
5 $$1 - 1151 T + p^{8} T^{2}$$
7 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
13 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
17 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
19 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
23 $$1 + 531793 T + p^{8} T^{2}$$
29 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
31 $$1 + 1541233 T + p^{8} T^{2}$$
37 $$1 - 716447 T + p^{8} T^{2}$$
41 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
43 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
47 $$1 + 6080638 T + p^{8} T^{2}$$
53 $$1 + 15265438 T + p^{8} T^{2}$$
59 $$1 + 4101553 T + p^{8} T^{2}$$
61 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
67 $$1 - 19806767 T + p^{8} T^{2}$$
71 $$1 - 7043087 T + p^{8} T^{2}$$
73 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
79 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
83 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
89 $$1 + 84100993 T + p^{8} T^{2}$$
97 $$1 + 81155713 T + p^{8} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$