Properties

Label 2-531-59.58-c4-0-61
Degree $2$
Conductor $531$
Sign $-0.00986 + 0.999i$
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  + 2.33i·2-s + 10.5·4-s − 30.0·5-s − 41.7·7-s + 62.0i·8-s − 70.2i·10-s + 117. i·11-s + 118. i·13-s − 97.6i·14-s + 23.2·16-s − 263.·17-s + 373.·19-s − 316.·20-s − 274.·22-s + 503. i·23-s + ⋯
L(s)  = 1  + 0.584i·2-s + 0.657·4-s − 1.20·5-s − 0.851·7-s + 0.969i·8-s − 0.702i·10-s + 0.970i·11-s + 0.700i·13-s − 0.498i·14-s + 0.0906·16-s − 0.913·17-s + 1.03·19-s − 0.790·20-s − 0.567·22-s + 0.951i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00986 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.00986 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.00986 + 0.999i$
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: \((2,\ 531,\ (\ :2),\ -0.00986 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.07955323401\)
\(L(\frac12)\) \(\approx\) \(0.07955323401\)
\(L(3)\) 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 \( 1 + (-34.3 + 3.48e3i)T \)
good2 \( 1 - 2.33iT - 16T^{2} \)
5 \( 1 + 30.0T + 625T^{2} \)
7 \( 1 + 41.7T + 2.40e3T^{2} \)
11 \( 1 - 117. iT - 1.46e4T^{2} \)
13 \( 1 - 118. iT - 2.85e4T^{2} \)
17 \( 1 + 263.T + 8.35e4T^{2} \)
19 \( 1 - 373.T + 1.30e5T^{2} \)
23 \( 1 - 503. iT - 2.79e5T^{2} \)
29 \( 1 - 549.T + 7.07e5T^{2} \)
31 \( 1 + 450. iT - 9.23e5T^{2} \)
37 \( 1 + 1.43e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.44e3T + 2.82e6T^{2} \)
43 \( 1 + 3.20e3iT - 3.41e6T^{2} \)
47 \( 1 + 967. iT - 4.87e6T^{2} \)
53 \( 1 + 1.66e3T + 7.89e6T^{2} \)
61 \( 1 - 1.71e3iT - 1.38e7T^{2} \)
67 \( 1 + 5.06e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.15e3T + 2.54e7T^{2} \)
73 \( 1 - 4.17e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.99e3T + 3.89e7T^{2} \)
83 \( 1 - 2.93e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.87e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.34e3iT - 8.85e7T^{2} \)
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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

−9.962446196479609730823617609770, −9.028110827030173829570067097126, −7.943570380177930606290155715366, −7.15748281198929235804427174888, −6.71426961307366504021616998470, −5.46132493550763765238298086020, −4.26838315394102448266956892363, −3.23334273267573930631696181094, −1.91665769960627153869428000715, −0.02221630745705846989186185961, 1.04730659179422944270979353750, 2.91327109731193202584471769114, 3.29962759263859527492706781291, 4.52929773665238910680211637450, 6.06106019249060997298661054967, 6.82069239663771014700124399201, 7.79721305126387382727188678020, 8.631689890195554838177530934311, 9.827510770107375663874153751892, 10.62570677657877787395662586811

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