# 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 − 19.1·5-s + 9·9-s − 40.5·11-s + 50.4·13-s − 57.4·15-s − 51.9·17-s + 33.1·19-s − 62.8·23-s + 241.·25-s + 27·27-s + 129.·29-s − 242.·31-s − 121.·33-s − 389.·37-s + 151.·39-s − 470.·41-s + 125.·43-s − 172.·45-s + 386.·47-s − 155.·51-s − 611.·53-s + 777.·55-s + 99.3·57-s − 226.·59-s + 725.·61-s − 966.·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.71·5-s + 0.333·9-s − 1.11·11-s + 1.07·13-s − 0.988·15-s − 0.740·17-s + 0.400·19-s − 0.569·23-s + 1.93·25-s + 0.192·27-s + 0.831·29-s − 1.40·31-s − 0.642·33-s − 1.73·37-s + 0.621·39-s − 1.79·41-s + 0.443·43-s − 0.570·45-s + 1.19·47-s − 0.427·51-s − 1.58·53-s + 1.90·55-s + 0.230·57-s − 0.499·59-s + 1.52·61-s − 1.84·65-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

 $$L(2)$$ $$\approx$$ $$1.089570633$$ $$L(\frac12)$$ $$\approx$$ $$1.089570633$$ $$L(\frac{5}{2})$$ 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 + 19.1T + 125T^{2}$$
11 $$1 + 40.5T + 1.33e3T^{2}$$
13 $$1 - 50.4T + 2.19e3T^{2}$$
17 $$1 + 51.9T + 4.91e3T^{2}$$
19 $$1 - 33.1T + 6.85e3T^{2}$$
23 $$1 + 62.8T + 1.21e4T^{2}$$
29 $$1 - 129.T + 2.43e4T^{2}$$
31 $$1 + 242.T + 2.97e4T^{2}$$
37 $$1 + 389.T + 5.06e4T^{2}$$
41 $$1 + 470.T + 6.89e4T^{2}$$
43 $$1 - 125.T + 7.95e4T^{2}$$
47 $$1 - 386.T + 1.03e5T^{2}$$
53 $$1 + 611.T + 1.48e5T^{2}$$
59 $$1 + 226.T + 2.05e5T^{2}$$
61 $$1 - 725.T + 2.26e5T^{2}$$
67 $$1 + 1.04e3T + 3.00e5T^{2}$$
71 $$1 + 169.T + 3.57e5T^{2}$$
73 $$1 + 381.T + 3.89e5T^{2}$$
79 $$1 - 1.16e3T + 4.93e5T^{2}$$
83 $$1 - 808.T + 5.71e5T^{2}$$
89 $$1 + 319.T + 7.04e5T^{2}$$
97 $$1 - 1.13e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$