Properties

Label 2-531-1.1-c7-0-36
Degree $2$
Conductor $531$
Sign $1$
Analytic cond. $165.876$
Root an. cond. $12.8793$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.7·2-s − 12.1·4-s − 451.·5-s − 504.·7-s + 1.50e3·8-s + 4.86e3·10-s + 8.07e3·11-s − 4.67e3·13-s + 5.43e3·14-s − 1.46e4·16-s + 1.34e4·17-s − 1.13e4·19-s + 5.49e3·20-s − 8.68e4·22-s + 6.64e4·23-s + 1.26e5·25-s + 5.03e4·26-s + 6.13e3·28-s + 1.74e5·29-s + 6.24e4·31-s − 3.50e4·32-s − 1.45e5·34-s + 2.28e5·35-s + 2.77e5·37-s + 1.22e5·38-s − 6.81e5·40-s − 5.29e5·41-s + ⋯
L(s)  = 1  − 0.951·2-s − 0.0949·4-s − 1.61·5-s − 0.556·7-s + 1.04·8-s + 1.53·10-s + 1.82·11-s − 0.590·13-s + 0.529·14-s − 0.895·16-s + 0.666·17-s − 0.380·19-s + 0.153·20-s − 1.73·22-s + 1.13·23-s + 1.61·25-s + 0.561·26-s + 0.0528·28-s + 1.32·29-s + 0.376·31-s − 0.189·32-s − 0.633·34-s + 0.899·35-s + 0.900·37-s + 0.361·38-s − 1.68·40-s − 1.20·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $1$
Analytic conductor: \(165.876\)
Root analytic conductor: \(12.8793\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.7552545269\)
\(L(\frac12)\) \(\approx\) \(0.7552545269\)
\(L(\frac{9}{2})\) 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 \( 1 - 2.05e5T \)
good2 \( 1 + 10.7T + 128T^{2} \)
5 \( 1 + 451.T + 7.81e4T^{2} \)
7 \( 1 + 504.T + 8.23e5T^{2} \)
11 \( 1 - 8.07e3T + 1.94e7T^{2} \)
13 \( 1 + 4.67e3T + 6.27e7T^{2} \)
17 \( 1 - 1.34e4T + 4.10e8T^{2} \)
19 \( 1 + 1.13e4T + 8.93e8T^{2} \)
23 \( 1 - 6.64e4T + 3.40e9T^{2} \)
29 \( 1 - 1.74e5T + 1.72e10T^{2} \)
31 \( 1 - 6.24e4T + 2.75e10T^{2} \)
37 \( 1 - 2.77e5T + 9.49e10T^{2} \)
41 \( 1 + 5.29e5T + 1.94e11T^{2} \)
43 \( 1 - 6.75e5T + 2.71e11T^{2} \)
47 \( 1 + 1.75e5T + 5.06e11T^{2} \)
53 \( 1 - 3.13e5T + 1.17e12T^{2} \)
61 \( 1 + 3.01e6T + 3.14e12T^{2} \)
67 \( 1 + 8.85e5T + 6.06e12T^{2} \)
71 \( 1 - 1.15e6T + 9.09e12T^{2} \)
73 \( 1 - 3.34e4T + 1.10e13T^{2} \)
79 \( 1 + 6.94e5T + 1.92e13T^{2} \)
83 \( 1 - 6.72e5T + 2.71e13T^{2} \)
89 \( 1 + 7.67e6T + 4.42e13T^{2} \)
97 \( 1 - 4.89e6T + 8.07e13T^{2} \)
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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

−9.493823320795331860124943691633, −8.843176294174679851966829658771, −8.047097111083515940336865168548, −7.22883598150130881643508087291, −6.49411369668523316349577189127, −4.73240808886345491389832516737, −4.05593742147713795927120572011, −3.08947043013686246643827966540, −1.25394845255747029777790641318, −0.51703408605922632415201744884, 0.51703408605922632415201744884, 1.25394845255747029777790641318, 3.08947043013686246643827966540, 4.05593742147713795927120572011, 4.73240808886345491389832516737, 6.49411369668523316349577189127, 7.22883598150130881643508087291, 8.047097111083515940336865168548, 8.843176294174679851966829658771, 9.493823320795331860124943691633

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