Properties

Label 2-531-1.1-c3-0-9
Degree $2$
Conductor $531$
Sign $1$
Analytic cond. $31.3300$
Root an. cond. $5.59732$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.17·2-s − 3.28·4-s − 9.58·5-s + 14.1·7-s + 24.5·8-s + 20.8·10-s − 19.1·11-s + 15.5·13-s − 30.7·14-s − 26.9·16-s − 100.·17-s + 74.3·19-s + 31.4·20-s + 41.5·22-s − 98.9·23-s − 33.2·25-s − 33.7·26-s − 46.5·28-s − 194.·29-s + 52.9·31-s − 137.·32-s + 218.·34-s − 135.·35-s − 212.·37-s − 161.·38-s − 234.·40-s + 395.·41-s + ⋯
L(s)  = 1  − 0.767·2-s − 0.410·4-s − 0.856·5-s + 0.764·7-s + 1.08·8-s + 0.657·10-s − 0.524·11-s + 0.331·13-s − 0.586·14-s − 0.420·16-s − 1.43·17-s + 0.897·19-s + 0.351·20-s + 0.402·22-s − 0.896·23-s − 0.265·25-s − 0.254·26-s − 0.313·28-s − 1.24·29-s + 0.306·31-s − 0.760·32-s + 1.10·34-s − 0.655·35-s − 0.945·37-s − 0.689·38-s − 0.928·40-s + 1.50·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $1$
Analytic conductor: \(31.3300\)
Root analytic conductor: \(5.59732\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7274212328\)
\(L(\frac12)\) \(\approx\) \(0.7274212328\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 - 59T \)
good2 \( 1 + 2.17T + 8T^{2} \)
5 \( 1 + 9.58T + 125T^{2} \)
7 \( 1 - 14.1T + 343T^{2} \)
11 \( 1 + 19.1T + 1.33e3T^{2} \)
13 \( 1 - 15.5T + 2.19e3T^{2} \)
17 \( 1 + 100.T + 4.91e3T^{2} \)
19 \( 1 - 74.3T + 6.85e3T^{2} \)
23 \( 1 + 98.9T + 1.21e4T^{2} \)
29 \( 1 + 194.T + 2.43e4T^{2} \)
31 \( 1 - 52.9T + 2.97e4T^{2} \)
37 \( 1 + 212.T + 5.06e4T^{2} \)
41 \( 1 - 395.T + 6.89e4T^{2} \)
43 \( 1 - 305.T + 7.95e4T^{2} \)
47 \( 1 - 630.T + 1.03e5T^{2} \)
53 \( 1 - 109.T + 1.48e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + 100.T + 3.00e5T^{2} \)
71 \( 1 + 263.T + 3.57e5T^{2} \)
73 \( 1 + 296.T + 3.89e5T^{2} \)
79 \( 1 - 626.T + 4.93e5T^{2} \)
83 \( 1 - 7.08T + 5.71e5T^{2} \)
89 \( 1 - 132.T + 7.04e5T^{2} \)
97 \( 1 - 1.47e3T + 9.12e5T^{2} \)
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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

−10.49452184855703936413409038854, −9.348833188171388466428978366572, −8.663231880042975895362944983757, −7.79103536238153792922905340822, −7.34299761242873969265351054364, −5.73115767095098912299568948622, −4.59540138514117634833337266535, −3.85042345154651054474684140492, −2.06265471250329851408014089616, −0.58595551638384297048090289261, 0.58595551638384297048090289261, 2.06265471250329851408014089616, 3.85042345154651054474684140492, 4.59540138514117634833337266535, 5.73115767095098912299568948622, 7.34299761242873969265351054364, 7.79103536238153792922905340822, 8.663231880042975895362944983757, 9.348833188171388466428978366572, 10.49452184855703936413409038854

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