# Properties

 Label 2-2352-1.1-c3-0-73 Degree $2$ Conductor $2352$ Sign $-1$ Analytic cond. $138.772$ Root an. cond. $11.7801$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3·3-s − 20.8·5-s + 9·9-s − 15.1·11-s − 2.16·13-s − 62.5·15-s + 119.·17-s − 33.5·19-s − 0.651·23-s + 309.·25-s + 27·27-s − 163.·29-s − 223.·31-s − 45.4·33-s + 168.·37-s − 6.48·39-s + 323.·41-s − 221.·43-s − 187.·45-s + 508.·47-s + 358.·51-s − 176.·53-s + 315.·55-s − 100.·57-s + 454.·59-s − 38.6·61-s + 45.0·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.86·5-s + 0.333·9-s − 0.415·11-s − 0.0461·13-s − 1.07·15-s + 1.70·17-s − 0.404·19-s − 0.00590·23-s + 2.47·25-s + 0.192·27-s − 1.04·29-s − 1.29·31-s − 0.239·33-s + 0.748·37-s − 0.0266·39-s + 1.23·41-s − 0.785·43-s − 0.621·45-s + 1.57·47-s + 0.983·51-s − 0.457·53-s + 0.774·55-s − 0.233·57-s + 1.00·59-s − 0.0811·61-s + 0.0860·65-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: $-1$ Analytic conductor: $$138.772$$ Root analytic conductor: $$11.7801$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes 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 + 20.8T + 125T^{2}$$
11 $$1 + 15.1T + 1.33e3T^{2}$$
13 $$1 + 2.16T + 2.19e3T^{2}$$
17 $$1 - 119.T + 4.91e3T^{2}$$
19 $$1 + 33.5T + 6.85e3T^{2}$$
23 $$1 + 0.651T + 1.21e4T^{2}$$
29 $$1 + 163.T + 2.43e4T^{2}$$
31 $$1 + 223.T + 2.97e4T^{2}$$
37 $$1 - 168.T + 5.06e4T^{2}$$
41 $$1 - 323.T + 6.89e4T^{2}$$
43 $$1 + 221.T + 7.95e4T^{2}$$
47 $$1 - 508.T + 1.03e5T^{2}$$
53 $$1 + 176.T + 1.48e5T^{2}$$
59 $$1 - 454.T + 2.05e5T^{2}$$
61 $$1 + 38.6T + 2.26e5T^{2}$$
67 $$1 + 141.T + 3.00e5T^{2}$$
71 $$1 + 602.T + 3.57e5T^{2}$$
73 $$1 - 1.10e3T + 3.89e5T^{2}$$
79 $$1 - 116.T + 4.93e5T^{2}$$
83 $$1 + 568.T + 5.71e5T^{2}$$
89 $$1 - 383.T + 7.04e5T^{2}$$
97 $$1 + 334.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$