# Properties

 Degree $2$ Conductor $6724$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.0950·3-s + 1.17·5-s − 3.14·7-s − 2.99·9-s + 1.67·11-s − 6.63·13-s − 0.111·15-s − 5.16·17-s + 4.72·19-s + 0.299·21-s − 8.82·23-s − 3.63·25-s + 0.569·27-s + 1.80·29-s − 1.65·31-s − 0.159·33-s − 3.68·35-s − 1.99·37-s + 0.630·39-s + 1.46·43-s − 3.50·45-s + 8.53·47-s + 2.89·49-s + 0.490·51-s + 9.35·53-s + 1.96·55-s − 0.449·57-s + ⋯
 L(s)  = 1 − 0.0549·3-s + 0.523·5-s − 1.18·7-s − 0.996·9-s + 0.505·11-s − 1.83·13-s − 0.0287·15-s − 1.25·17-s + 1.08·19-s + 0.0652·21-s − 1.83·23-s − 0.726·25-s + 0.109·27-s + 0.336·29-s − 0.298·31-s − 0.0277·33-s − 0.622·35-s − 0.327·37-s + 0.100·39-s + 0.224·43-s − 0.521·45-s + 1.24·47-s + 0.413·49-s + 0.0687·51-s + 1.28·53-s + 0.264·55-s − 0.0595·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6724$$    =    $$2^{2} \cdot 41^{2}$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{6724} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 6724,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8081021128$$ $$L(\frac12)$$ $$\approx$$ $$0.8081021128$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
41 $$1$$
good3 $$1 + 0.0950T + 3T^{2}$$
5 $$1 - 1.17T + 5T^{2}$$
7 $$1 + 3.14T + 7T^{2}$$
11 $$1 - 1.67T + 11T^{2}$$
13 $$1 + 6.63T + 13T^{2}$$
17 $$1 + 5.16T + 17T^{2}$$
19 $$1 - 4.72T + 19T^{2}$$
23 $$1 + 8.82T + 23T^{2}$$
29 $$1 - 1.80T + 29T^{2}$$
31 $$1 + 1.65T + 31T^{2}$$
37 $$1 + 1.99T + 37T^{2}$$
43 $$1 - 1.46T + 43T^{2}$$
47 $$1 - 8.53T + 47T^{2}$$
53 $$1 - 9.35T + 53T^{2}$$
59 $$1 - 8.82T + 59T^{2}$$
61 $$1 - 12.6T + 61T^{2}$$
67 $$1 + 9.67T + 67T^{2}$$
71 $$1 - 0.776T + 71T^{2}$$
73 $$1 - 8.33T + 73T^{2}$$
79 $$1 + 0.915T + 79T^{2}$$
83 $$1 + 10.0T + 83T^{2}$$
89 $$1 - 6.44T + 89T^{2}$$
97 $$1 - 8.32T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$