Properties

Label 2-531-1.1-c7-0-150
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  + 15.0·2-s + 98.4·4-s − 159.·5-s + 980.·7-s − 445.·8-s − 2.40e3·10-s + 3.39e3·11-s − 7.16e3·13-s + 1.47e4·14-s − 1.92e4·16-s + 1.22e4·17-s + 1.95e4·19-s − 1.57e4·20-s + 5.10e4·22-s − 1.03e5·23-s − 5.25e4·25-s − 1.07e5·26-s + 9.64e4·28-s + 1.69e5·29-s − 3.77e4·31-s − 2.33e5·32-s + 1.84e5·34-s − 1.56e5·35-s − 2.95e4·37-s + 2.94e5·38-s + 7.12e4·40-s − 3.75e5·41-s + ⋯
L(s)  = 1  + 1.32·2-s + 0.768·4-s − 0.572·5-s + 1.08·7-s − 0.307·8-s − 0.760·10-s + 0.768·11-s − 0.905·13-s + 1.43·14-s − 1.17·16-s + 0.604·17-s + 0.654·19-s − 0.439·20-s + 1.02·22-s − 1.76·23-s − 0.672·25-s − 1.20·26-s + 0.830·28-s + 1.28·29-s − 0.227·31-s − 1.25·32-s + 0.803·34-s − 0.617·35-s − 0.0958·37-s + 0.870·38-s + 0.175·40-s − 0.849·41-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 \( 1 - 15.0T + 128T^{2} \)
5 \( 1 + 159.T + 7.81e4T^{2} \)
7 \( 1 - 980.T + 8.23e5T^{2} \)
11 \( 1 - 3.39e3T + 1.94e7T^{2} \)
13 \( 1 + 7.16e3T + 6.27e7T^{2} \)
17 \( 1 - 1.22e4T + 4.10e8T^{2} \)
19 \( 1 - 1.95e4T + 8.93e8T^{2} \)
23 \( 1 + 1.03e5T + 3.40e9T^{2} \)
29 \( 1 - 1.69e5T + 1.72e10T^{2} \)
31 \( 1 + 3.77e4T + 2.75e10T^{2} \)
37 \( 1 + 2.95e4T + 9.49e10T^{2} \)
41 \( 1 + 3.75e5T + 1.94e11T^{2} \)
43 \( 1 - 7.19e5T + 2.71e11T^{2} \)
47 \( 1 - 3.17e5T + 5.06e11T^{2} \)
53 \( 1 + 2.90e5T + 1.17e12T^{2} \)
61 \( 1 + 2.85e6T + 3.14e12T^{2} \)
67 \( 1 - 1.89e6T + 6.06e12T^{2} \)
71 \( 1 + 3.57e6T + 9.09e12T^{2} \)
73 \( 1 + 2.76e6T + 1.10e13T^{2} \)
79 \( 1 + 4.56e6T + 1.92e13T^{2} \)
83 \( 1 - 6.61e5T + 2.71e13T^{2} \)
89 \( 1 - 5.31e6T + 4.42e13T^{2} \)
97 \( 1 + 1.84e6T + 8.07e13T^{2} \)
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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

−9.259410418901815360742634971440, −8.128981671374583335993996892588, −7.41405553230353855215190682182, −6.23464887914067040344351105284, −5.33688828048038911198497231339, −4.46842443504332065991258514018, −3.85090577020874408327404268640, −2.69331605719864040662205289897, −1.49184665751187722813609008011, 0, 1.49184665751187722813609008011, 2.69331605719864040662205289897, 3.85090577020874408327404268640, 4.46842443504332065991258514018, 5.33688828048038911198497231339, 6.23464887914067040344351105284, 7.41405553230353855215190682182, 8.128981671374583335993996892588, 9.259410418901815360742634971440

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