Properties

Label 2-539-1.1-c3-0-97
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  + 2.73·2-s + 7.92·3-s − 0.535·4-s − 14.8·5-s + 21.6·6-s − 23.3·8-s + 35.8·9-s − 40.5·10-s − 11·11-s − 4.24·12-s − 5.35·13-s − 117.·15-s − 59.4·16-s + 41.2·17-s + 97.9·18-s − 139.·19-s + 7.96·20-s − 30.0·22-s − 111.·23-s − 184.·24-s + 95.7·25-s − 14.6·26-s + 70.2·27-s − 24.9·29-s − 321.·30-s − 31.4·31-s + 24.2·32-s + ⋯
L(s)  = 1  + 0.965·2-s + 1.52·3-s − 0.0669·4-s − 1.32·5-s + 1.47·6-s − 1.03·8-s + 1.32·9-s − 1.28·10-s − 0.301·11-s − 0.102·12-s − 0.114·13-s − 2.02·15-s − 0.928·16-s + 0.588·17-s + 1.28·18-s − 1.68·19-s + 0.0890·20-s − 0.291·22-s − 1.00·23-s − 1.57·24-s + 0.765·25-s − 0.110·26-s + 0.500·27-s − 0.160·29-s − 1.95·30-s − 0.182·31-s + 0.133·32-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 \( 1 - 2.73T + 8T^{2} \)
3 \( 1 - 7.92T + 27T^{2} \)
5 \( 1 + 14.8T + 125T^{2} \)
13 \( 1 + 5.35T + 2.19e3T^{2} \)
17 \( 1 - 41.2T + 4.91e3T^{2} \)
19 \( 1 + 139.T + 6.85e3T^{2} \)
23 \( 1 + 111.T + 1.21e4T^{2} \)
29 \( 1 + 24.9T + 2.43e4T^{2} \)
31 \( 1 + 31.4T + 2.97e4T^{2} \)
37 \( 1 - 13.1T + 5.06e4T^{2} \)
41 \( 1 + 261.T + 6.89e4T^{2} \)
43 \( 1 + 57.7T + 7.95e4T^{2} \)
47 \( 1 - 343.T + 1.03e5T^{2} \)
53 \( 1 + 342.T + 1.48e5T^{2} \)
59 \( 1 + 88.3T + 2.05e5T^{2} \)
61 \( 1 + 738.T + 2.26e5T^{2} \)
67 \( 1 - 342.T + 3.00e5T^{2} \)
71 \( 1 + 207.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 - 1.29e3T + 4.93e5T^{2} \)
83 \( 1 + 441.T + 5.71e5T^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + 1.34e3T + 9.12e5T^{2} \)
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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

−9.807914654316438045463829338871, −8.809421479825941995080082368838, −8.204192816959429673320366388741, −7.52993170768753087772705693121, −6.24886501379630788260314473550, −4.80093132741622055213586054978, −3.92858771523133254829491865070, −3.41282771992919602999633646117, −2.26688064799501805895340432730, 0, 2.26688064799501805895340432730, 3.41282771992919602999633646117, 3.92858771523133254829491865070, 4.80093132741622055213586054978, 6.24886501379630788260314473550, 7.52993170768753087772705693121, 8.204192816959429673320366388741, 8.809421479825941995080082368838, 9.807914654316438045463829338871

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