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Properties

Label 2-531-3.2-c4-0-30
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  + 5.75i·2-s − 17.0·4-s + 6.47i·5-s − 90.4·7-s − 6.19i·8-s − 37.2·10-s − 40.5i·11-s − 265.·13-s − 520. i·14-s − 237.·16-s + 87.1i·17-s + 108.·19-s − 110. i·20-s + 232.·22-s + 671. i·23-s + ⋯
L(s)  = 1  + 1.43i·2-s − 1.06·4-s + 0.258i·5-s − 1.84·7-s − 0.0968i·8-s − 0.372·10-s − 0.334i·11-s − 1.57·13-s − 2.65i·14-s − 0.928·16-s + 0.301i·17-s + 0.301·19-s − 0.276i·20-s + 0.481·22-s + 1.27i·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

\(L(\frac{5}{2})\) \(\approx\) \(0.5345746467\)
\(L(\frac12)\) \(\approx\) \(0.5345746467\)
\(L(3)\) 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 \( 1 - 5.75iT - 16T^{2} \)
5 \( 1 - 6.47iT - 625T^{2} \)
7 \( 1 + 90.4T + 2.40e3T^{2} \)
11 \( 1 + 40.5iT - 1.46e4T^{2} \)
13 \( 1 + 265.T + 2.85e4T^{2} \)
17 \( 1 - 87.1iT - 8.35e4T^{2} \)
19 \( 1 - 108.T + 1.30e5T^{2} \)
23 \( 1 - 671. iT - 2.79e5T^{2} \)
29 \( 1 + 742. iT - 7.07e5T^{2} \)
31 \( 1 + 1.19e3T + 9.23e5T^{2} \)
37 \( 1 - 760.T + 1.87e6T^{2} \)
41 \( 1 - 3.26e3iT - 2.82e6T^{2} \)
43 \( 1 + 178.T + 3.41e6T^{2} \)
47 \( 1 + 3.59e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.11e3iT - 7.89e6T^{2} \)
61 \( 1 - 5.27e3T + 1.38e7T^{2} \)
67 \( 1 + 869.T + 2.01e7T^{2} \)
71 \( 1 + 18.5iT - 2.54e7T^{2} \)
73 \( 1 - 2.48e3T + 2.83e7T^{2} \)
79 \( 1 - 9.97e3T + 3.89e7T^{2} \)
83 \( 1 - 5.04e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.64e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.26e4T + 8.85e7T^{2} \)
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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

−9.754052183302976410745290684158, −9.561493257940045943138612595260, −8.318523709394493389293531058814, −7.28194770097434457448256123439, −6.79832668660581622084297857729, −5.93170217878770129742930284839, −5.09343868849751552785973343616, −3.65003094373415210251309215945, −2.55155733342294241964127846273, −0.20348133651892745044915827570, 0.69794224796377512630926915163, 2.34654247470000332328610482980, 3.01599857026832092421191708714, 4.07255151754134612909084720757, 5.18883833469272130963960529148, 6.62716890626823855986232584837, 7.29133455565911537507883042107, 9.038207272280867419372034356498, 9.444616052508162284168622271129, 10.22534878937040083778230082601

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