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Properties

Label 2-531-59.58-c4-0-19
Degree $2$
Conductor $531$
Sign $-0.867 - 0.497i$
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  + 1.07i·2-s + 14.8·4-s − 26.1·5-s + 49.1·7-s + 33.2i·8-s − 28.1i·10-s + 190. i·11-s + 198. i·13-s + 52.9i·14-s + 201.·16-s + 81.6·17-s − 590.·19-s − 387.·20-s − 205.·22-s − 808. i·23-s + ⋯
L(s)  = 1  + 0.269i·2-s + 0.927·4-s − 1.04·5-s + 1.00·7-s + 0.519i·8-s − 0.281i·10-s + 1.57i·11-s + 1.17i·13-s + 0.270i·14-s + 0.787·16-s + 0.282·17-s − 1.63·19-s − 0.969·20-s − 0.424·22-s − 1.52i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.497i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.867 - 0.497i$
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: \((2,\ 531,\ (\ :2),\ -0.867 - 0.497i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.524978052\)
\(L(\frac12)\) \(\approx\) \(1.524978052\)
\(L(3)\) 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 \( 1 + (-3.01e3 - 1.73e3i)T \)
good2 \( 1 - 1.07iT - 16T^{2} \)
5 \( 1 + 26.1T + 625T^{2} \)
7 \( 1 - 49.1T + 2.40e3T^{2} \)
11 \( 1 - 190. iT - 1.46e4T^{2} \)
13 \( 1 - 198. iT - 2.85e4T^{2} \)
17 \( 1 - 81.6T + 8.35e4T^{2} \)
19 \( 1 + 590.T + 1.30e5T^{2} \)
23 \( 1 + 808. iT - 2.79e5T^{2} \)
29 \( 1 - 714.T + 7.07e5T^{2} \)
31 \( 1 + 1.37e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.10e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.13e3T + 2.82e6T^{2} \)
43 \( 1 - 349. iT - 3.41e6T^{2} \)
47 \( 1 - 2.25e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.73e3T + 7.89e6T^{2} \)
61 \( 1 + 1.50e3iT - 1.38e7T^{2} \)
67 \( 1 + 233. iT - 2.01e7T^{2} \)
71 \( 1 + 2.66e3T + 2.54e7T^{2} \)
73 \( 1 - 9.12e3iT - 2.83e7T^{2} \)
79 \( 1 + 8.28e3T + 3.89e7T^{2} \)
83 \( 1 + 7.47e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.50e4iT - 8.85e7T^{2} \)
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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

−10.78798859715871763378015135124, −9.853047266758032324099591138815, −8.417554286945792004697527082282, −7.944247049539694175403863949812, −6.98259479860855604273796167271, −6.35997884628176490992001375565, −4.66896412307951799319319855998, −4.27430831269823916101318727119, −2.49035912785542278499592781764, −1.60409790822633123910738861318, 0.36249968851624885972137102593, 1.59413505709824299157701952619, 3.05897026404342969732838547643, 3.78423440052625469929358164603, 5.22218408313367887065977270665, 6.16097697082323911840756361808, 7.35356906757075080015004482167, 8.096604625766380590427768588286, 8.630590265769988029478620939259, 10.30795572457803309917064550383

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