# Properties

 Degree $2$ Conductor $1$ Sign $1$ Motivic weight $11$ Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 24·2-s + 252·3-s − 1.47e3·4-s + 4.83e3·5-s − 6.04e3·6-s − 1.67e4·7-s + 8.44e4·8-s − 1.13e5·9-s − 1.15e5·10-s + 5.34e5·11-s − 3.70e5·12-s − 5.77e5·13-s + 4.01e5·14-s + 1.21e6·15-s + 9.87e5·16-s − 6.90e6·17-s + 2.72e6·18-s + 1.06e7·19-s − 7.10e6·20-s − 4.21e6·21-s − 1.28e7·22-s + 1.86e7·23-s + 2.12e7·24-s − 2.54e7·25-s + 1.38e7·26-s − 7.32e7·27-s + 2.46e7·28-s + ⋯
 L(s)  = 1 − 0.530·2-s + 0.598·3-s − 0.718·4-s + 0.691·5-s − 0.317·6-s − 0.376·7-s + 0.911·8-s − 0.641·9-s − 0.366·10-s + 1.00·11-s − 0.430·12-s − 0.431·13-s + 0.199·14-s + 0.413·15-s + 0.235·16-s − 1.17·17-s + 0.340·18-s + 0.987·19-s − 0.496·20-s − 0.225·21-s − 0.530·22-s + 0.603·23-s + 0.545·24-s − 0.522·25-s + 0.228·26-s − 0.982·27-s + 0.270·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+11/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1$$ Sign: $1$ Motivic weight: $$11$$ Character: $\chi_{1} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1,\ (\ :11/2),\ 1)$$

## Particular Values

 $$L(6)$$ $$\approx$$ $$0.792122$$ $$L(\frac12)$$ $$\approx$$ $$0.792122$$ $$L(\frac{13}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
good2 $$1 + 3 p^{3} T + p^{11} T^{2}$$
3 $$1 - 28 p^{2} T + p^{11} T^{2}$$
5 $$1 - 966 p T + p^{11} T^{2}$$
7 $$1 + 2392 p T + p^{11} T^{2}$$
11 $$1 - 534612 T + p^{11} T^{2}$$
13 $$1 + 577738 T + p^{11} T^{2}$$
17 $$1 + 6905934 T + p^{11} T^{2}$$
19 $$1 - 10661420 T + p^{11} T^{2}$$
23 $$1 - 18643272 T + p^{11} T^{2}$$
29 $$1 - 128406630 T + p^{11} T^{2}$$
31 $$1 + 52843168 T + p^{11} T^{2}$$
37 $$1 + 182213314 T + p^{11} T^{2}$$
41 $$1 - 308120442 T + p^{11} T^{2}$$
43 $$1 + 17125708 T + p^{11} T^{2}$$
47 $$1 - 2687348496 T + p^{11} T^{2}$$
53 $$1 + 1596055698 T + p^{11} T^{2}$$
59 $$1 + 5189203740 T + p^{11} T^{2}$$
61 $$1 - 6956478662 T + p^{11} T^{2}$$
67 $$1 + 15481826884 T + p^{11} T^{2}$$
71 $$1 - 9791485272 T + p^{11} T^{2}$$
73 $$1 - 1463791322 T + p^{11} T^{2}$$
79 $$1 - 38116845680 T + p^{11} T^{2}$$
83 $$1 + 29335099668 T + p^{11} T^{2}$$
89 $$1 + 24992917110 T + p^{11} T^{2}$$
97 $$1 - 75013568546 T + p^{11} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$