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Properties

Label 2-531-3.2-c4-0-10
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  + 4.10i·2-s − 0.830·4-s − 6.63i·5-s + 3.66·7-s + 62.2i·8-s + 27.2·10-s + 44.2i·11-s + 32.0·13-s + 15.0i·14-s − 268.·16-s − 340. i·17-s − 603.·19-s + 5.50i·20-s − 181.·22-s + 800. i·23-s + ⋯
L(s)  = 1  + 1.02i·2-s − 0.0519·4-s − 0.265i·5-s + 0.0747·7-s + 0.972i·8-s + 0.272·10-s + 0.365i·11-s + 0.189·13-s + 0.0766i·14-s − 1.04·16-s − 1.17i·17-s − 1.67·19-s + 0.0137i·20-s − 0.375·22-s + 1.51i·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

\(L(\frac{5}{2})\) \(\approx\) \(0.9257884848\)
\(L(\frac12)\) \(\approx\) \(0.9257884848\)
\(L(3)\) 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 \( 1 - 4.10iT - 16T^{2} \)
5 \( 1 + 6.63iT - 625T^{2} \)
7 \( 1 - 3.66T + 2.40e3T^{2} \)
11 \( 1 - 44.2iT - 1.46e4T^{2} \)
13 \( 1 - 32.0T + 2.85e4T^{2} \)
17 \( 1 + 340. iT - 8.35e4T^{2} \)
19 \( 1 + 603.T + 1.30e5T^{2} \)
23 \( 1 - 800. iT - 2.79e5T^{2} \)
29 \( 1 - 239. iT - 7.07e5T^{2} \)
31 \( 1 - 356.T + 9.23e5T^{2} \)
37 \( 1 + 154.T + 1.87e6T^{2} \)
41 \( 1 - 383. iT - 2.82e6T^{2} \)
43 \( 1 + 2.58e3T + 3.41e6T^{2} \)
47 \( 1 - 2.54e3iT - 4.87e6T^{2} \)
53 \( 1 + 502. iT - 7.89e6T^{2} \)
61 \( 1 + 6.41e3T + 1.38e7T^{2} \)
67 \( 1 - 1.82e3T + 2.01e7T^{2} \)
71 \( 1 - 1.83e3iT - 2.54e7T^{2} \)
73 \( 1 + 125.T + 2.83e7T^{2} \)
79 \( 1 + 1.52e3T + 3.89e7T^{2} \)
83 \( 1 - 1.21e4iT - 4.74e7T^{2} \)
89 \( 1 + 1.44e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.43e4T + 8.85e7T^{2} \)
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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

−10.85620377798274873116218551373, −9.671682028932170344061444782296, −8.763014277193726254775137855335, −7.973062526612297110695016822429, −7.07556758376042802840536077772, −6.36551572122847882558361320437, −5.29787429643241787976176728768, −4.51338785880362042096960471738, −2.92486621813370179632177220071, −1.60704150229985254067811827442, 0.21649591090774783825171632201, 1.61150565057013814468771308439, 2.59624462202808858312880253102, 3.66413622346972974373427180157, 4.61462612178798376625295371620, 6.24729634942631770899845510062, 6.71612851629679281290875144813, 8.183429106481084134629058337670, 8.864859665030575674209406743647, 10.19326208326076482536133477661

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