# Properties

 Label 2-539-1.1-c3-0-54 Degree $2$ Conductor $539$ Sign $-1$ Analytic cond. $31.8020$ Root an. cond. $5.63932$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 0.732·2-s − 5.92·3-s − 7.46·4-s + 12.8·5-s + 4.33·6-s + 11.3·8-s + 8.14·9-s − 9.41·10-s − 11·11-s + 44.2·12-s − 74.6·13-s − 76.2·15-s + 51.4·16-s + 82.7·17-s − 5.96·18-s + 67.9·19-s − 95.9·20-s + 8.05·22-s + 13.3·23-s − 67.1·24-s + 40.2·25-s + 54.6·26-s + 111.·27-s + 168.·29-s + 55.7·30-s + 65.4·31-s − 128.·32-s + ⋯
 L(s)  = 1 − 0.258·2-s − 1.14·3-s − 0.933·4-s + 1.14·5-s + 0.295·6-s + 0.500·8-s + 0.301·9-s − 0.297·10-s − 0.301·11-s + 1.06·12-s − 1.59·13-s − 1.31·15-s + 0.803·16-s + 1.18·17-s − 0.0780·18-s + 0.820·19-s − 1.07·20-s + 0.0780·22-s + 0.121·23-s − 0.570·24-s + 0.322·25-s + 0.412·26-s + 0.796·27-s + 1.08·29-s + 0.339·30-s + 0.379·31-s − 0.708·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$539$$    =    $$7^{2} \cdot 11$$ Sign: $-1$ Analytic conductor: $$31.8020$$ Root analytic conductor: $$5.63932$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{539} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 539,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad7 $$1$$
11 $$1 + 11T$$
good2 $$1 + 0.732T + 8T^{2}$$
3 $$1 + 5.92T + 27T^{2}$$
5 $$1 - 12.8T + 125T^{2}$$
13 $$1 + 74.6T + 2.19e3T^{2}$$
17 $$1 - 82.7T + 4.91e3T^{2}$$
19 $$1 - 67.9T + 6.85e3T^{2}$$
23 $$1 - 13.3T + 1.21e4T^{2}$$
29 $$1 - 168.T + 2.43e4T^{2}$$
31 $$1 - 65.4T + 2.97e4T^{2}$$
37 $$1 - 40.8T + 5.06e4T^{2}$$
41 $$1 + 274.T + 6.89e4T^{2}$$
43 $$1 + 2.28T + 7.95e4T^{2}$$
47 $$1 + 71.8T + 1.03e5T^{2}$$
53 $$1 + 149.T + 1.48e5T^{2}$$
59 $$1 + 545.T + 2.05e5T^{2}$$
61 $$1 + 101.T + 2.26e5T^{2}$$
67 $$1 - 411.T + 3.00e5T^{2}$$
71 $$1 + 470.T + 3.57e5T^{2}$$
73 $$1 + 610.T + 3.89e5T^{2}$$
79 $$1 + 978.T + 4.93e5T^{2}$$
83 $$1 + 26.1T + 5.71e5T^{2}$$
89 $$1 - 352.T + 7.04e5T^{2}$$
97 $$1 + 847.T + 9.12e5T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.971991675506078997338983093270, −9.454726767681092905591339790584, −8.226300289574639172701729561315, −7.20122426204243107397606490682, −5.99425873795175285840569123791, −5.27460625594674344609724228187, −4.72968351477440373709047311640, −2.91691763428118188016738661083, −1.25927469338755089467574721630, 0, 1.25927469338755089467574721630, 2.91691763428118188016738661083, 4.72968351477440373709047311640, 5.27460625594674344609724228187, 5.99425873795175285840569123791, 7.20122426204243107397606490682, 8.226300289574639172701729561315, 9.454726767681092905591339790584, 9.971991675506078997338983093270