# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 3·3-s − 15·5-s + 9·9-s + 9·11-s − 88·13-s − 45·15-s − 84·17-s − 104·19-s + 84·23-s + 100·25-s + 27·27-s + 51·29-s − 185·31-s + 27·33-s + 44·37-s − 264·39-s − 168·41-s − 326·43-s − 135·45-s + 138·47-s − 252·51-s + 639·53-s − 135·55-s − 312·57-s − 159·59-s + 722·61-s + 1.32e3·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.34·5-s + 1/3·9-s + 0.246·11-s − 1.87·13-s − 0.774·15-s − 1.19·17-s − 1.25·19-s + 0.761·23-s + 4/5·25-s + 0.192·27-s + 0.326·29-s − 1.07·31-s + 0.142·33-s + 0.195·37-s − 1.08·39-s − 0.639·41-s − 1.15·43-s − 0.447·45-s + 0.428·47-s − 0.691·51-s + 1.65·53-s − 0.330·55-s − 0.725·57-s − 0.350·59-s + 1.51·61-s + 2.51·65-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

 $$L(2)$$ $$\approx$$ $$0.8373211712$$ $$L(\frac12)$$ $$\approx$$ $$0.8373211712$$ $$L(\frac{5}{2})$$ 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 - p T$$
7 $$1$$
good5 $$1 + 3 p T + p^{3} T^{2}$$
11 $$1 - 9 T + p^{3} T^{2}$$
13 $$1 + 88 T + p^{3} T^{2}$$
17 $$1 + 84 T + p^{3} T^{2}$$
19 $$1 + 104 T + p^{3} T^{2}$$
23 $$1 - 84 T + p^{3} T^{2}$$
29 $$1 - 51 T + p^{3} T^{2}$$
31 $$1 + 185 T + p^{3} T^{2}$$
37 $$1 - 44 T + p^{3} T^{2}$$
41 $$1 + 168 T + p^{3} T^{2}$$
43 $$1 + 326 T + p^{3} T^{2}$$
47 $$1 - 138 T + p^{3} T^{2}$$
53 $$1 - 639 T + p^{3} T^{2}$$
59 $$1 + 159 T + p^{3} T^{2}$$
61 $$1 - 722 T + p^{3} T^{2}$$
67 $$1 - 166 T + p^{3} T^{2}$$
71 $$1 + 1086 T + p^{3} T^{2}$$
73 $$1 - 218 T + p^{3} T^{2}$$
79 $$1 - 583 T + p^{3} T^{2}$$
83 $$1 - 597 T + p^{3} T^{2}$$
89 $$1 + 1038 T + p^{3} T^{2}$$
97 $$1 + 169 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$