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Properties

Label 2-531-59.58-c4-0-37
Degree $2$
Conductor $531$
Sign $0.999 + 0.00476i$
Analytic cond. $54.8894$
Root an. cond. $7.40874$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.10i·2-s − 0.836·4-s + 39.6·5-s − 85.8·7-s − 62.2i·8-s − 162. i·10-s + 68.0i·11-s + 254. i·13-s + 352. i·14-s − 268.·16-s − 229.·17-s + 425.·19-s − 33.1·20-s + 279.·22-s + 954. i·23-s + ⋯
L(s)  = 1  − 1.02i·2-s − 0.0522·4-s + 1.58·5-s − 1.75·7-s − 0.972i·8-s − 1.62i·10-s + 0.562i·11-s + 1.50i·13-s + 1.79i·14-s − 1.04·16-s − 0.794·17-s + 1.17·19-s − 0.0828·20-s + 0.577·22-s + 1.80i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00476i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 + 0.00476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.999 + 0.00476i$
Analytic conductor: \(54.8894\)
Root analytic conductor: \(7.40874\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :2),\ 0.999 + 0.00476i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.191462393\)
\(L(\frac12)\) \(\approx\) \(2.191462393\)
\(L(3)\) 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 \( 1 + (3.48e3 + 16.5i)T \)
good2 \( 1 + 4.10iT - 16T^{2} \)
5 \( 1 - 39.6T + 625T^{2} \)
7 \( 1 + 85.8T + 2.40e3T^{2} \)
11 \( 1 - 68.0iT - 1.46e4T^{2} \)
13 \( 1 - 254. iT - 2.85e4T^{2} \)
17 \( 1 + 229.T + 8.35e4T^{2} \)
19 \( 1 - 425.T + 1.30e5T^{2} \)
23 \( 1 - 954. iT - 2.79e5T^{2} \)
29 \( 1 - 396.T + 7.07e5T^{2} \)
31 \( 1 + 1.42e3iT - 9.23e5T^{2} \)
37 \( 1 - 996. iT - 1.87e6T^{2} \)
41 \( 1 - 1.02e3T + 2.82e6T^{2} \)
43 \( 1 - 2.58e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.47e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.15e3T + 7.89e6T^{2} \)
61 \( 1 - 2.17e3iT - 1.38e7T^{2} \)
67 \( 1 - 667. iT - 2.01e7T^{2} \)
71 \( 1 - 2.58e3T + 2.54e7T^{2} \)
73 \( 1 - 1.20e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.24e3T + 3.89e7T^{2} \)
83 \( 1 - 6.11e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.20e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.69e4iT - 8.85e7T^{2} \)
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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

−9.941197110566588901540668900535, −9.465524677555011062015984701826, −9.393097868269827624218585287150, −7.18809893074310004026657714851, −6.51579383869887433831876221715, −5.79386991933911848236010302808, −4.23699821597875590468972578939, −3.05161536456190284786006067134, −2.22644189411722975467183734701, −1.24362755445639401818992689564, 0.54487759787998576833743399876, 2.41083382620786856991429511466, 3.16731879488047514491045556340, 5.16999013632259256675020810511, 5.87879327747552244587766971406, 6.44982458831330353915861141278, 7.16939995225567614701903040594, 8.552058738368116485315901651312, 9.195621722694081862556598618194, 10.28262201074050136591892939519

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