# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s + 3.89·5-s + 9·9-s + 61.3·11-s − 53.6·13-s − 11.6·15-s − 32.1·17-s − 55.7·19-s + 94.6·23-s − 109.·25-s − 27·27-s + 138.·29-s + 132.·31-s − 184.·33-s + 149.·37-s + 161.·39-s − 427.·41-s − 437.·43-s + 35.0·45-s + 57.0·47-s + 96.3·51-s − 263.·53-s + 239.·55-s + 167.·57-s + 451.·59-s − 579.·61-s − 209.·65-s + ⋯
 L(s)  = 1 − 0.577·3-s + 0.348·5-s + 0.333·9-s + 1.68·11-s − 1.14·13-s − 0.201·15-s − 0.457·17-s − 0.673·19-s + 0.857·23-s − 0.878·25-s − 0.192·27-s + 0.884·29-s + 0.768·31-s − 0.971·33-s + 0.662·37-s + 0.661·39-s − 1.62·41-s − 1.55·43-s + 0.116·45-s + 0.176·47-s + 0.264·51-s − 0.683·53-s + 0.586·55-s + 0.388·57-s + 0.996·59-s − 1.21·61-s − 0.399·65-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: $$1$$ Selberg data: $$(2,\ 2352,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 3.89T + 125T^{2}$$
11 $$1 - 61.3T + 1.33e3T^{2}$$
13 $$1 + 53.6T + 2.19e3T^{2}$$
17 $$1 + 32.1T + 4.91e3T^{2}$$
19 $$1 + 55.7T + 6.85e3T^{2}$$
23 $$1 - 94.6T + 1.21e4T^{2}$$
29 $$1 - 138.T + 2.43e4T^{2}$$
31 $$1 - 132.T + 2.97e4T^{2}$$
37 $$1 - 149.T + 5.06e4T^{2}$$
41 $$1 + 427.T + 6.89e4T^{2}$$
43 $$1 + 437.T + 7.95e4T^{2}$$
47 $$1 - 57.0T + 1.03e5T^{2}$$
53 $$1 + 263.T + 1.48e5T^{2}$$
59 $$1 - 451.T + 2.05e5T^{2}$$
61 $$1 + 579.T + 2.26e5T^{2}$$
67 $$1 + 309.T + 3.00e5T^{2}$$
71 $$1 - 1.05e3T + 3.57e5T^{2}$$
73 $$1 + 1.19e3T + 3.89e5T^{2}$$
79 $$1 + 1.31e3T + 4.93e5T^{2}$$
83 $$1 - 1.19e3T + 5.71e5T^{2}$$
89 $$1 - 233.T + 7.04e5T^{2}$$
97 $$1 - 1.60e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$