Properties

Label 2-531-59.58-c4-0-88
Degree $2$
Conductor $531$
Sign $-0.471 + 0.882i$
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  + 0.389i·2-s + 15.8·4-s + 17.9·5-s − 47.2·7-s + 12.3i·8-s + 7.00i·10-s − 197. i·11-s + 176. i·13-s − 18.3i·14-s + 248.·16-s − 486.·17-s + 56.2·19-s + 285.·20-s + 76.7·22-s − 848. i·23-s + ⋯
L(s)  = 1  + 0.0972i·2-s + 0.990·4-s + 0.719·5-s − 0.964·7-s + 0.193i·8-s + 0.0700i·10-s − 1.63i·11-s + 1.04i·13-s − 0.0938i·14-s + 0.971·16-s − 1.68·17-s + 0.155·19-s + 0.713·20-s + 0.158·22-s − 1.60i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.882i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.471 + 0.882i$
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: \((2,\ 531,\ (\ :2),\ -0.471 + 0.882i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.456473453\)
\(L(\frac12)\) \(\approx\) \(1.456473453\)
\(L(3)\) 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 \( 1 + (-1.64e3 + 3.07e3i)T \)
good2 \( 1 - 0.389iT - 16T^{2} \)
5 \( 1 - 17.9T + 625T^{2} \)
7 \( 1 + 47.2T + 2.40e3T^{2} \)
11 \( 1 + 197. iT - 1.46e4T^{2} \)
13 \( 1 - 176. iT - 2.85e4T^{2} \)
17 \( 1 + 486.T + 8.35e4T^{2} \)
19 \( 1 - 56.2T + 1.30e5T^{2} \)
23 \( 1 + 848. iT - 2.79e5T^{2} \)
29 \( 1 + 275.T + 7.07e5T^{2} \)
31 \( 1 + 843. iT - 9.23e5T^{2} \)
37 \( 1 + 1.85e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.47e3T + 2.82e6T^{2} \)
43 \( 1 - 2.75e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.43e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.11e3T + 7.89e6T^{2} \)
61 \( 1 - 115. iT - 1.38e7T^{2} \)
67 \( 1 - 49.5iT - 2.01e7T^{2} \)
71 \( 1 - 2.08e3T + 2.54e7T^{2} \)
73 \( 1 - 209. iT - 2.83e7T^{2} \)
79 \( 1 + 6.54e3T + 3.89e7T^{2} \)
83 \( 1 - 6.80e3iT - 4.74e7T^{2} \)
89 \( 1 - 9.11e3iT - 6.27e7T^{2} \)
97 \( 1 - 2.94e3iT - 8.85e7T^{2} \)
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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

−9.954323933722637670741021101323, −9.092632170704081539276080774087, −8.251699805981362686955564975829, −6.79837195319661881673868920582, −6.41929539495608904311325539531, −5.65441287919658241122548102246, −4.05110880800414749389197150689, −2.82472630897100231838087601414, −1.97095849656835540040022291993, −0.31452494382940946077546868910, 1.58815905476157319142309553906, 2.46731540205499090743325363514, 3.56182274774005834512339702730, 5.06154838188947864884830301011, 6.08599625761252245474140111943, 6.88786652510547699996311922224, 7.54087152386126736788811554261, 8.951858902790220728436021084125, 9.991692958825703008624428119314, 10.21215229407629391504779378509

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