Properties

Label 2-531-1.1-c7-0-91
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.9·2-s + 193.·4-s − 1.22·5-s − 719.·7-s − 1.18e3·8-s + 21.9·10-s − 1.11e3·11-s + 8.28e3·13-s + 1.29e4·14-s − 3.58e3·16-s − 4.83e3·17-s − 3.16e4·19-s − 237.·20-s + 1.99e4·22-s + 4.35e4·23-s − 7.81e4·25-s − 1.48e5·26-s − 1.39e5·28-s + 2.32e5·29-s − 8.53e4·31-s + 2.15e5·32-s + 8.67e4·34-s + 879.·35-s − 5.27e4·37-s + 5.67e5·38-s + 1.44e3·40-s + 5.89e5·41-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.51·4-s − 0.00437·5-s − 0.792·7-s − 0.817·8-s + 0.00694·10-s − 0.251·11-s + 1.04·13-s + 1.25·14-s − 0.218·16-s − 0.238·17-s − 1.05·19-s − 0.00663·20-s + 0.399·22-s + 0.746·23-s − 0.999·25-s − 1.65·26-s − 1.20·28-s + 1.77·29-s − 0.514·31-s + 1.16·32-s + 0.378·34-s + 0.00346·35-s − 0.171·37-s + 1.67·38-s + 0.00357·40-s + 1.33·41-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: \(1\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 17.9T + 128T^{2} \)
5 \( 1 + 1.22T + 7.81e4T^{2} \)
7 \( 1 + 719.T + 8.23e5T^{2} \)
11 \( 1 + 1.11e3T + 1.94e7T^{2} \)
13 \( 1 - 8.28e3T + 6.27e7T^{2} \)
17 \( 1 + 4.83e3T + 4.10e8T^{2} \)
19 \( 1 + 3.16e4T + 8.93e8T^{2} \)
23 \( 1 - 4.35e4T + 3.40e9T^{2} \)
29 \( 1 - 2.32e5T + 1.72e10T^{2} \)
31 \( 1 + 8.53e4T + 2.75e10T^{2} \)
37 \( 1 + 5.27e4T + 9.49e10T^{2} \)
41 \( 1 - 5.89e5T + 1.94e11T^{2} \)
43 \( 1 - 3.67e5T + 2.71e11T^{2} \)
47 \( 1 + 7.39e5T + 5.06e11T^{2} \)
53 \( 1 + 1.53e6T + 1.17e12T^{2} \)
61 \( 1 - 3.25e5T + 3.14e12T^{2} \)
67 \( 1 + 1.72e6T + 6.06e12T^{2} \)
71 \( 1 - 4.29e6T + 9.09e12T^{2} \)
73 \( 1 + 1.55e6T + 1.10e13T^{2} \)
79 \( 1 + 3.39e6T + 1.92e13T^{2} \)
83 \( 1 - 7.04e6T + 2.71e13T^{2} \)
89 \( 1 + 1.14e7T + 4.42e13T^{2} \)
97 \( 1 - 1.36e7T + 8.07e13T^{2} \)
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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

−9.209535858545779336574183979162, −8.522037511642843556914608230593, −7.75091435342293318104477755792, −6.67609328299477646290131685178, −6.09412299254353106762173067666, −4.48493314066391438567971198985, −3.17304338575547649716658866744, −2.04767650883440137915100891365, −0.933251052791349761453853512215, 0, 0.933251052791349761453853512215, 2.04767650883440137915100891365, 3.17304338575547649716658866744, 4.48493314066391438567971198985, 6.09412299254353106762173067666, 6.67609328299477646290131685178, 7.75091435342293318104477755792, 8.522037511642843556914608230593, 9.209535858545779336574183979162

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