# Properties

 Label 2-4840-1.1-c1-0-100 Degree $2$ Conductor $4840$ Sign $-1$ Analytic cond. $38.6475$ Root an. cond. $6.21671$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.33·3-s − 5-s − 0.399·7-s + 2.43·9-s − 2.43·13-s − 2.33·15-s − 3.63·19-s − 0.932·21-s − 1.02·23-s + 25-s − 1.30·27-s − 4·29-s + 3.23·31-s + 0.399·35-s − 4.87·37-s − 5.68·39-s − 0.0922·41-s − 7.14·43-s − 2.43·45-s + 9.21·47-s − 6.84·49-s + 9.10·53-s − 8.48·57-s − 7.68·59-s − 6.57·61-s − 0.975·63-s + 2.43·65-s + ⋯
 L(s)  = 1 + 1.34·3-s − 0.447·5-s − 0.151·7-s + 0.813·9-s − 0.676·13-s − 0.602·15-s − 0.835·19-s − 0.203·21-s − 0.213·23-s + 0.200·25-s − 0.251·27-s − 0.742·29-s + 0.581·31-s + 0.0675·35-s − 0.802·37-s − 0.911·39-s − 0.0144·41-s − 1.08·43-s − 0.363·45-s + 1.34·47-s − 0.977·49-s + 1.25·53-s − 1.12·57-s − 1.00·59-s − 0.841·61-s − 0.122·63-s + 0.302·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4840$$    =    $$2^{3} \cdot 5 \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$38.6475$$ Root analytic conductor: $$6.21671$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4840,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + T$$
11 $$1$$
good3 $$1 - 2.33T + 3T^{2}$$
7 $$1 + 0.399T + 7T^{2}$$
13 $$1 + 2.43T + 13T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + 3.63T + 19T^{2}$$
23 $$1 + 1.02T + 23T^{2}$$
29 $$1 + 4T + 29T^{2}$$
31 $$1 - 3.23T + 31T^{2}$$
37 $$1 + 4.87T + 37T^{2}$$
41 $$1 + 0.0922T + 41T^{2}$$
43 $$1 + 7.14T + 43T^{2}$$
47 $$1 - 9.21T + 47T^{2}$$
53 $$1 - 9.10T + 53T^{2}$$
59 $$1 + 7.68T + 59T^{2}$$
61 $$1 + 6.57T + 61T^{2}$$
67 $$1 + 11.8T + 67T^{2}$$
71 $$1 - 5.28T + 71T^{2}$$
73 $$1 + 8T + 73T^{2}$$
79 $$1 - 1.67T + 79T^{2}$$
83 $$1 + 16.3T + 83T^{2}$$
89 $$1 - 10.2T + 89T^{2}$$
97 $$1 - 11.5T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$