Properties

Label 2-531-3.2-c4-0-42
Degree $2$
Conductor $531$
Sign $-0.816 + 0.577i$
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  − 5.40i·2-s − 13.2·4-s + 5.33i·5-s − 40.2·7-s − 14.9i·8-s + 28.8·10-s + 236. i·11-s + 228.·13-s + 217. i·14-s − 292.·16-s − 143. i·17-s − 422.·19-s − 70.6i·20-s + 1.27e3·22-s − 486. i·23-s + ⋯
L(s)  = 1  − 1.35i·2-s − 0.827·4-s + 0.213i·5-s − 0.821·7-s − 0.233i·8-s + 0.288·10-s + 1.95i·11-s + 1.35·13-s + 1.11i·14-s − 1.14·16-s − 0.495i·17-s − 1.17·19-s − 0.176i·20-s + 2.63·22-s − 0.918i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(54.8894\)
Root analytic conductor: \(7.40874\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :2),\ -0.816 + 0.577i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.622864848\)
\(L(\frac12)\) \(\approx\) \(1.622864848\)
\(L(3)\) 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 \( 1 - 453. iT \)
good2 \( 1 + 5.40iT - 16T^{2} \)
5 \( 1 - 5.33iT - 625T^{2} \)
7 \( 1 + 40.2T + 2.40e3T^{2} \)
11 \( 1 - 236. iT - 1.46e4T^{2} \)
13 \( 1 - 228.T + 2.85e4T^{2} \)
17 \( 1 + 143. iT - 8.35e4T^{2} \)
19 \( 1 + 422.T + 1.30e5T^{2} \)
23 \( 1 + 486. iT - 2.79e5T^{2} \)
29 \( 1 + 464. iT - 7.07e5T^{2} \)
31 \( 1 - 1.10e3T + 9.23e5T^{2} \)
37 \( 1 + 60.3T + 1.87e6T^{2} \)
41 \( 1 + 404. iT - 2.82e6T^{2} \)
43 \( 1 - 1.54e3T + 3.41e6T^{2} \)
47 \( 1 + 1.12e3iT - 4.87e6T^{2} \)
53 \( 1 + 5.19e3iT - 7.89e6T^{2} \)
61 \( 1 + 126.T + 1.38e7T^{2} \)
67 \( 1 - 5.06e3T + 2.01e7T^{2} \)
71 \( 1 + 3.20e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.56e3T + 2.83e7T^{2} \)
79 \( 1 - 71.8T + 3.89e7T^{2} \)
83 \( 1 + 4.14e3iT - 4.74e7T^{2} \)
89 \( 1 + 7.28e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.10e4T + 8.85e7T^{2} \)
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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

−10.08942970161495216195106131487, −9.399851924673973465017684643058, −8.401770140698098257480315855139, −6.91765642980037655688903852131, −6.43576127913233089521317464443, −4.68975764747875425576770414142, −3.89333136064062724777559323089, −2.75476029013549012709330001690, −1.88056704995904762259306376091, −0.50546674178132231193833836464, 0.975917042975538816801649377351, 2.96198360199834956789910908065, 4.03786961190387217237117834893, 5.47694137565113610462087542794, 6.19348017901942187693462680329, 6.63321838913569255941471668952, 8.051284724222524078000748617689, 8.550990754438057334653423282909, 9.243267393461456799796284347202, 10.76004027017702725068471282879

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