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 $1$

Origins

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Normalization:  

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

Graph of the $Z$-function along the critical line