Properties

Label 2-531-1.1-c3-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.26·2-s − 6.38·4-s − 14.8·5-s + 22.4·7-s − 18.2·8-s − 18.8·10-s + 70.2·11-s − 0.125·13-s + 28.5·14-s + 27.9·16-s − 57.2·17-s + 40.8·19-s + 95.0·20-s + 89.1·22-s − 190.·23-s + 96.4·25-s − 0.159·26-s − 143.·28-s − 133.·29-s + 129.·31-s + 181.·32-s − 72.6·34-s − 334.·35-s − 364.·37-s + 51.8·38-s + 271.·40-s − 195.·41-s + ⋯
L(s)  = 1  + 0.448·2-s − 0.798·4-s − 1.33·5-s + 1.21·7-s − 0.807·8-s − 0.597·10-s + 1.92·11-s − 0.00267·13-s + 0.544·14-s + 0.436·16-s − 0.817·17-s + 0.492·19-s + 1.06·20-s + 0.863·22-s − 1.72·23-s + 0.771·25-s − 0.00119·26-s − 0.969·28-s − 0.854·29-s + 0.747·31-s + 1.00·32-s − 0.366·34-s − 1.61·35-s − 1.61·37-s + 0.221·38-s + 1.07·40-s − 0.745·41-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: \(1\)
Selberg data: \((2,\ 531,\ (\ :3/2),\ -1)\)

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)$
bad3 \( 1 \)
59 \( 1 + 59T \)
good2 \( 1 - 1.26T + 8T^{2} \)
5 \( 1 + 14.8T + 125T^{2} \)
7 \( 1 - 22.4T + 343T^{2} \)
11 \( 1 - 70.2T + 1.33e3T^{2} \)
13 \( 1 + 0.125T + 2.19e3T^{2} \)
17 \( 1 + 57.2T + 4.91e3T^{2} \)
19 \( 1 - 40.8T + 6.85e3T^{2} \)
23 \( 1 + 190.T + 1.21e4T^{2} \)
29 \( 1 + 133.T + 2.43e4T^{2} \)
31 \( 1 - 129.T + 2.97e4T^{2} \)
37 \( 1 + 364.T + 5.06e4T^{2} \)
41 \( 1 + 195.T + 6.89e4T^{2} \)
43 \( 1 - 31.6T + 7.95e4T^{2} \)
47 \( 1 + 479.T + 1.03e5T^{2} \)
53 \( 1 - 402.T + 1.48e5T^{2} \)
61 \( 1 + 209.T + 2.26e5T^{2} \)
67 \( 1 + 455.T + 3.00e5T^{2} \)
71 \( 1 + 203.T + 3.57e5T^{2} \)
73 \( 1 + 177.T + 3.89e5T^{2} \)
79 \( 1 + 491.T + 4.93e5T^{2} \)
83 \( 1 - 717.T + 5.71e5T^{2} \)
89 \( 1 + 1.30e3T + 7.04e5T^{2} \)
97 \( 1 - 538.T + 9.12e5T^{2} \)
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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

−9.930095342063241973946476732346, −8.827298654578787132838738871098, −8.350507956284348445405893959056, −7.37826104055766617641746583917, −6.22755830808620544267207449427, −4.92877657723712254076159566987, −4.14916151427918170366543512791, −3.61315644693119236256655834351, −1.53554027591694275079747291984, 0, 1.53554027591694275079747291984, 3.61315644693119236256655834351, 4.14916151427918170366543512791, 4.92877657723712254076159566987, 6.22755830808620544267207449427, 7.37826104055766617641746583917, 8.350507956284348445405893959056, 8.827298654578787132838738871098, 9.930095342063241973946476732346

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