# 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 + 16.6·5-s + 9·9-s + 71.7·11-s + 65.3·13-s + 49.9·15-s − 90.4·17-s + 163.·19-s − 79.2·23-s + 152.·25-s + 27·27-s − 43.2·29-s + 135.·31-s + 215.·33-s + 270.·37-s + 196.·39-s − 152.·41-s + 177.·43-s + 149.·45-s − 45.6·47-s − 271.·51-s − 158.·53-s + 1.19e3·55-s + 491.·57-s − 391.·59-s − 551.·61-s + 1.08e3·65-s + ⋯
 L(s)  = 1 + 0.577·3-s + 1.48·5-s + 0.333·9-s + 1.96·11-s + 1.39·13-s + 0.860·15-s − 1.29·17-s + 1.97·19-s − 0.718·23-s + 1.21·25-s + 0.192·27-s − 0.277·29-s + 0.785·31-s + 1.13·33-s + 1.20·37-s + 0.805·39-s − 0.579·41-s + 0.630·43-s + 0.496·45-s − 0.141·47-s − 0.744·51-s − 0.410·53-s + 2.93·55-s + 1.14·57-s − 0.864·59-s − 1.15·61-s + 2.07·65-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

 $$L(2)$$ $$\approx$$ $$5.328369992$$ $$L(\frac12)$$ $$\approx$$ $$5.328369992$$ $$L(\frac{5}{2})$$ 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 - 16.6T + 125T^{2}$$
11 $$1 - 71.7T + 1.33e3T^{2}$$
13 $$1 - 65.3T + 2.19e3T^{2}$$
17 $$1 + 90.4T + 4.91e3T^{2}$$
19 $$1 - 163.T + 6.85e3T^{2}$$
23 $$1 + 79.2T + 1.21e4T^{2}$$
29 $$1 + 43.2T + 2.43e4T^{2}$$
31 $$1 - 135.T + 2.97e4T^{2}$$
37 $$1 - 270.T + 5.06e4T^{2}$$
41 $$1 + 152.T + 6.89e4T^{2}$$
43 $$1 - 177.T + 7.95e4T^{2}$$
47 $$1 + 45.6T + 1.03e5T^{2}$$
53 $$1 + 158.T + 1.48e5T^{2}$$
59 $$1 + 391.T + 2.05e5T^{2}$$
61 $$1 + 551.T + 2.26e5T^{2}$$
67 $$1 + 458.T + 3.00e5T^{2}$$
71 $$1 - 486.T + 3.57e5T^{2}$$
73 $$1 + 574.T + 3.89e5T^{2}$$
79 $$1 - 668.T + 4.93e5T^{2}$$
83 $$1 - 76.2T + 5.71e5T^{2}$$
89 $$1 + 1.36e3T + 7.04e5T^{2}$$
97 $$1 + 242.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$