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Properties

Label 2-531-3.2-c4-0-5
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  − 0.525i·2-s + 15.7·4-s + 43.0i·5-s − 78.9·7-s − 16.6i·8-s + 22.6·10-s + 138. i·11-s − 107.·13-s + 41.4i·14-s + 242.·16-s − 388. i·17-s + 70.8·19-s + 677. i·20-s + 72.8·22-s + 873. i·23-s + ⋯
L(s)  = 1  − 0.131i·2-s + 0.982·4-s + 1.72i·5-s − 1.61·7-s − 0.260i·8-s + 0.226·10-s + 1.14i·11-s − 0.636·13-s + 0.211i·14-s + 0.948·16-s − 1.34i·17-s + 0.196·19-s + 1.69i·20-s + 0.150·22-s + 1.65i·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

\(L(\frac{5}{2})\) \(\approx\) \(0.4081712861\)
\(L(\frac12)\) \(\approx\) \(0.4081712861\)
\(L(3)\) 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 \( 1 + 0.525iT - 16T^{2} \)
5 \( 1 - 43.0iT - 625T^{2} \)
7 \( 1 + 78.9T + 2.40e3T^{2} \)
11 \( 1 - 138. iT - 1.46e4T^{2} \)
13 \( 1 + 107.T + 2.85e4T^{2} \)
17 \( 1 + 388. iT - 8.35e4T^{2} \)
19 \( 1 - 70.8T + 1.30e5T^{2} \)
23 \( 1 - 873. iT - 2.79e5T^{2} \)
29 \( 1 - 16.7iT - 7.07e5T^{2} \)
31 \( 1 - 1.00e3T + 9.23e5T^{2} \)
37 \( 1 + 2.12e3T + 1.87e6T^{2} \)
41 \( 1 + 2.10e3iT - 2.82e6T^{2} \)
43 \( 1 + 3.08e3T + 3.41e6T^{2} \)
47 \( 1 + 1.17e3iT - 4.87e6T^{2} \)
53 \( 1 + 13.8iT - 7.89e6T^{2} \)
61 \( 1 - 3.86e3T + 1.38e7T^{2} \)
67 \( 1 + 330.T + 2.01e7T^{2} \)
71 \( 1 + 6.27e3iT - 2.54e7T^{2} \)
73 \( 1 - 928.T + 2.83e7T^{2} \)
79 \( 1 + 1.83e3T + 3.89e7T^{2} \)
83 \( 1 + 1.15e4iT - 4.74e7T^{2} \)
89 \( 1 - 1.03e4iT - 6.27e7T^{2} \)
97 \( 1 + 838.T + 8.85e7T^{2} \)
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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

−10.57636990853543623492823647893, −9.986096648579921680042018644493, −9.518639795291281031845214941869, −7.54858498971724348091471925715, −6.93622817689908335054543836685, −6.68526604038575783948771916133, −5.43591726688379745013199812681, −3.53734724201412790412296025179, −2.96303311240665601373373694081, −2.07689260653560956813405533931, 0.097993484458714877026485511271, 1.24245673090907531472219196298, 2.71980008924633533530402694427, 3.81970808471006651386115611965, 5.15667929937595587583399900119, 6.13062953961808711465332924877, 6.70083303137625050553388349542, 8.202890630075279444527069702018, 8.605850452992499437712383773990, 9.772250769735907449660756638530

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