Properties

Label 2-531-59.58-c4-0-78
Degree $2$
Conductor $531$
Sign $-0.736 + 0.676i$
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  − 4.85i·2-s − 7.60·4-s + 12.5·5-s + 50.4·7-s − 40.7i·8-s − 60.7i·10-s − 191. i·11-s − 0.640i·13-s − 245. i·14-s − 319.·16-s + 549.·17-s + 524.·19-s − 95.0·20-s − 929.·22-s + 543. i·23-s + ⋯
L(s)  = 1  − 1.21i·2-s − 0.475·4-s + 0.500·5-s + 1.03·7-s − 0.637i·8-s − 0.607i·10-s − 1.58i·11-s − 0.00378i·13-s − 1.25i·14-s − 1.24·16-s + 1.90·17-s + 1.45·19-s − 0.237·20-s − 1.91·22-s + 1.02i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.736 + 0.676i$
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: \((2,\ 531,\ (\ :2),\ -0.736 + 0.676i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.107006569\)
\(L(\frac12)\) \(\approx\) \(3.107006569\)
\(L(3)\) 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 \( 1 + (-2.56e3 + 2.35e3i)T \)
good2 \( 1 + 4.85iT - 16T^{2} \)
5 \( 1 - 12.5T + 625T^{2} \)
7 \( 1 - 50.4T + 2.40e3T^{2} \)
11 \( 1 + 191. iT - 1.46e4T^{2} \)
13 \( 1 + 0.640iT - 2.85e4T^{2} \)
17 \( 1 - 549.T + 8.35e4T^{2} \)
19 \( 1 - 524.T + 1.30e5T^{2} \)
23 \( 1 - 543. iT - 2.79e5T^{2} \)
29 \( 1 - 296.T + 7.07e5T^{2} \)
31 \( 1 - 1.07e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.16e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.12e3T + 2.82e6T^{2} \)
43 \( 1 - 1.52e3iT - 3.41e6T^{2} \)
47 \( 1 + 103. iT - 4.87e6T^{2} \)
53 \( 1 - 2.81e3T + 7.89e6T^{2} \)
61 \( 1 + 4.90e3iT - 1.38e7T^{2} \)
67 \( 1 + 5.85e3iT - 2.01e7T^{2} \)
71 \( 1 + 5.82e3T + 2.54e7T^{2} \)
73 \( 1 - 6.75e3iT - 2.83e7T^{2} \)
79 \( 1 + 9.33e3T + 3.89e7T^{2} \)
83 \( 1 + 1.83e3iT - 4.74e7T^{2} \)
89 \( 1 + 5.73e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.52e4iT - 8.85e7T^{2} \)
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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

−10.03395825682435781371763257685, −9.368647144355246139074753939173, −8.221455854642131371157583704471, −7.38992025260859897357069182467, −5.88914980623416365745061218027, −5.21399274924335142540654676964, −3.65462076203458523650228605678, −2.98265312723435675075945774414, −1.55342724430119648758375968912, −0.900710323041666575098444734945, 1.34828556719389393001832645286, 2.50918772200617040590506237965, 4.34498523306217374638413020341, 5.27285523148960599224905035656, 5.88657133257462375482778506548, 7.22793766658756290044657430237, 7.59051919507562890985392296281, 8.485442741833969610133219742343, 9.678727504216698417618121125514, 10.27300947933396910360646840890

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