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Properties

Label 2-531-59.58-c4-0-83
Degree $2$
Conductor $531$
Sign $0.586 + 0.810i$
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  + 6.10i·2-s − 21.2·4-s + 12.8·5-s + 4.61·7-s − 32.0i·8-s + 78.4i·10-s − 159. i·11-s + 148. i·13-s + 28.1i·14-s − 144.·16-s − 154.·17-s + 313.·19-s − 273.·20-s + 973.·22-s + 420. i·23-s + ⋯
L(s)  = 1  + 1.52i·2-s − 1.32·4-s + 0.513·5-s + 0.0942·7-s − 0.500i·8-s + 0.784i·10-s − 1.31i·11-s + 0.881i·13-s + 0.143i·14-s − 0.564·16-s − 0.534·17-s + 0.867·19-s − 0.682·20-s + 2.01·22-s + 0.794i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.586 + 0.810i$
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: \((2,\ 531,\ (\ :2),\ 0.586 + 0.810i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.2118090436\)
\(L(\frac12)\) \(\approx\) \(0.2118090436\)
\(L(3)\) 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 \( 1 + (2.04e3 + 2.82e3i)T \)
good2 \( 1 - 6.10iT - 16T^{2} \)
5 \( 1 - 12.8T + 625T^{2} \)
7 \( 1 - 4.61T + 2.40e3T^{2} \)
11 \( 1 + 159. iT - 1.46e4T^{2} \)
13 \( 1 - 148. iT - 2.85e4T^{2} \)
17 \( 1 + 154.T + 8.35e4T^{2} \)
19 \( 1 - 313.T + 1.30e5T^{2} \)
23 \( 1 - 420. iT - 2.79e5T^{2} \)
29 \( 1 + 1.42e3T + 7.07e5T^{2} \)
31 \( 1 + 1.10e3iT - 9.23e5T^{2} \)
37 \( 1 + 186. iT - 1.87e6T^{2} \)
41 \( 1 + 435.T + 2.82e6T^{2} \)
43 \( 1 - 2.36e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.36e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.94e3T + 7.89e6T^{2} \)
61 \( 1 - 1.73e3iT - 1.38e7T^{2} \)
67 \( 1 + 2.71e3iT - 2.01e7T^{2} \)
71 \( 1 + 5.43e3T + 2.54e7T^{2} \)
73 \( 1 + 2.07e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.07e4T + 3.89e7T^{2} \)
83 \( 1 + 1.09e4iT - 4.74e7T^{2} \)
89 \( 1 + 1.73e3iT - 6.27e7T^{2} \)
97 \( 1 - 3.59e3iT - 8.85e7T^{2} \)
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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

−9.578094676234808071757834009406, −9.139587678175867897448302668819, −8.075218245876274053581539514956, −7.39688899000257637533262816864, −6.27443089326615352247509569435, −5.81051853164161648732743130873, −4.81842490584111852180959960476, −3.53168990068116931784361585346, −1.85571689087231993846800631219, −0.05116023187739505437891822078, 1.41704023403277577633708094894, 2.25408085511365796628465439409, 3.32251902550852511743573589837, 4.45437270041073268450454489790, 5.40044497236885231555045092199, 6.76577402593064331606737069652, 7.81512951496158663806566760723, 9.119040347283559416815996819319, 9.715538830626093663189308744348, 10.43039908760746775075213336590

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