# Properties

 Degree $2$ Conductor $2352$ Sign $1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3·3-s + 19.8·5-s + 9·9-s − 23.9·11-s + 87.3·13-s + 59.6·15-s − 5.63·17-s − 64.8·19-s + 25.5·23-s + 270.·25-s + 27·27-s + 60.3·29-s + 122.·31-s − 71.8·33-s − 56.1·37-s + 262.·39-s − 299.·41-s + 501.·43-s + 179.·45-s − 305.·47-s − 16.9·51-s − 375.·53-s − 476.·55-s − 194.·57-s + 627.·59-s + 3.75·61-s + 1.73e3·65-s + ⋯
 L(s)  = 1 + 0.577·3-s + 1.77·5-s + 0.333·9-s − 0.656·11-s + 1.86·13-s + 1.02·15-s − 0.0804·17-s − 0.783·19-s + 0.232·23-s + 2.16·25-s + 0.192·27-s + 0.386·29-s + 0.710·31-s − 0.378·33-s − 0.249·37-s + 1.07·39-s − 1.14·41-s + 1.77·43-s + 0.593·45-s − 0.948·47-s − 0.0464·51-s − 0.972·53-s − 1.16·55-s − 0.452·57-s + 1.38·59-s + 0.00788·61-s + 3.31·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2352$$    =    $$2^{4} \cdot 3 \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$3$$ Character: $\chi_{2352} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2352,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$4.735833730$$ $$L(\frac12)$$ $$\approx$$ $$4.735833730$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - 3T$$
7 $$1$$
good5 $$1 - 19.8T + 125T^{2}$$
11 $$1 + 23.9T + 1.33e3T^{2}$$
13 $$1 - 87.3T + 2.19e3T^{2}$$
17 $$1 + 5.63T + 4.91e3T^{2}$$
19 $$1 + 64.8T + 6.85e3T^{2}$$
23 $$1 - 25.5T + 1.21e4T^{2}$$
29 $$1 - 60.3T + 2.43e4T^{2}$$
31 $$1 - 122.T + 2.97e4T^{2}$$
37 $$1 + 56.1T + 5.06e4T^{2}$$
41 $$1 + 299.T + 6.89e4T^{2}$$
43 $$1 - 501.T + 7.95e4T^{2}$$
47 $$1 + 305.T + 1.03e5T^{2}$$
53 $$1 + 375.T + 1.48e5T^{2}$$
59 $$1 - 627.T + 2.05e5T^{2}$$
61 $$1 - 3.75T + 2.26e5T^{2}$$
67 $$1 - 813.T + 3.00e5T^{2}$$
71 $$1 + 165.T + 3.57e5T^{2}$$
73 $$1 - 619.T + 3.89e5T^{2}$$
79 $$1 - 138.T + 4.93e5T^{2}$$
83 $$1 + 621.T + 5.71e5T^{2}$$
89 $$1 - 285.T + 7.04e5T^{2}$$
97 $$1 + 603.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$