Properties

Label 2-531-1.1-c7-0-144
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  − 19.6·2-s + 258.·4-s + 436.·5-s + 956.·7-s − 2.56e3·8-s − 8.58e3·10-s − 1.97e3·11-s − 5.30e3·13-s − 1.88e4·14-s + 1.72e4·16-s + 1.91e4·17-s − 2.14e4·19-s + 1.12e5·20-s + 3.87e4·22-s + 3.43e4·23-s + 1.12e5·25-s + 1.04e5·26-s + 2.47e5·28-s − 4.00e4·29-s − 9.54e4·31-s − 1.20e4·32-s − 3.77e5·34-s + 4.17e5·35-s − 2.69e5·37-s + 4.21e5·38-s − 1.11e6·40-s + 3.60e5·41-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.01·4-s + 1.56·5-s + 1.05·7-s − 1.76·8-s − 2.71·10-s − 0.446·11-s − 0.670·13-s − 1.83·14-s + 1.05·16-s + 0.947·17-s − 0.717·19-s + 3.15·20-s + 0.776·22-s + 0.588·23-s + 1.44·25-s + 1.16·26-s + 2.12·28-s − 0.304·29-s − 0.575·31-s − 0.0647·32-s − 1.64·34-s + 1.64·35-s − 0.873·37-s + 1.24·38-s − 2.76·40-s + 0.816·41-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 \( 1 + 19.6T + 128T^{2} \)
5 \( 1 - 436.T + 7.81e4T^{2} \)
7 \( 1 - 956.T + 8.23e5T^{2} \)
11 \( 1 + 1.97e3T + 1.94e7T^{2} \)
13 \( 1 + 5.30e3T + 6.27e7T^{2} \)
17 \( 1 - 1.91e4T + 4.10e8T^{2} \)
19 \( 1 + 2.14e4T + 8.93e8T^{2} \)
23 \( 1 - 3.43e4T + 3.40e9T^{2} \)
29 \( 1 + 4.00e4T + 1.72e10T^{2} \)
31 \( 1 + 9.54e4T + 2.75e10T^{2} \)
37 \( 1 + 2.69e5T + 9.49e10T^{2} \)
41 \( 1 - 3.60e5T + 1.94e11T^{2} \)
43 \( 1 + 9.29e5T + 2.71e11T^{2} \)
47 \( 1 + 5.40e4T + 5.06e11T^{2} \)
53 \( 1 + 6.80e5T + 1.17e12T^{2} \)
61 \( 1 + 1.27e6T + 3.14e12T^{2} \)
67 \( 1 - 6.30e5T + 6.06e12T^{2} \)
71 \( 1 + 2.05e6T + 9.09e12T^{2} \)
73 \( 1 + 5.36e6T + 1.10e13T^{2} \)
79 \( 1 - 4.57e6T + 1.92e13T^{2} \)
83 \( 1 + 9.49e6T + 2.71e13T^{2} \)
89 \( 1 - 2.45e6T + 4.42e13T^{2} \)
97 \( 1 + 1.41e7T + 8.07e13T^{2} \)
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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

−9.318562735172092708029798165633, −8.525990251244530057256478559619, −7.70923160573212043709722887436, −6.84563333487282297356757364960, −5.77414495256800856556121886620, −4.90948809355291715127213268689, −2.81648164372211644347165496125, −1.85192107895209792244282561895, −1.36755121563122431001990874951, 0, 1.36755121563122431001990874951, 1.85192107895209792244282561895, 2.81648164372211644347165496125, 4.90948809355291715127213268689, 5.77414495256800856556121886620, 6.84563333487282297356757364960, 7.70923160573212043709722887436, 8.525990251244530057256478559619, 9.318562735172092708029798165633

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