Properties

Label 2-539-1.1-c1-0-26
Degree $2$
Conductor $539$
Sign $-1$
Analytic cond. $4.30393$
Root an. cond. $2.07459$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 4-s + 2·5-s − 2·6-s − 3·8-s + 9-s + 2·10-s + 11-s + 2·12-s − 4·13-s − 4·15-s − 16-s − 4·17-s + 18-s − 2·20-s + 22-s − 4·23-s + 6·24-s − 25-s − 4·26-s + 4·27-s − 6·29-s − 4·30-s − 10·31-s + 5·32-s − 2·33-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.894·5-s − 0.816·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.10·13-s − 1.03·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.447·20-s + 0.213·22-s − 0.834·23-s + 1.22·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s − 1.11·29-s − 0.730·30-s − 1.79·31-s + 0.883·32-s − 0.348·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -1)\)

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)$
bad7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.50466724059560752904922488149, −9.519898606480484632296989725464, −8.920762498226765508675400216409, −7.37145602881524282201485228293, −6.25417326558836201502812232893, −5.62343675689709412781095073358, −4.94543315829774789302801165548, −3.86661484554997482031837898039, −2.20053354537635753960919690772, 0, 2.20053354537635753960919690772, 3.86661484554997482031837898039, 4.94543315829774789302801165548, 5.62343675689709412781095073358, 6.25417326558836201502812232893, 7.37145602881524282201485228293, 8.920762498226765508675400216409, 9.519898606480484632296989725464, 10.50466724059560752904922488149

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