Properties

Label 2-531-1.1-c7-0-102
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  + 12.3·2-s + 24.9·4-s + 477.·5-s + 1.34e3·7-s − 1.27e3·8-s + 5.91e3·10-s − 6.22e3·11-s + 1.09e4·13-s + 1.66e4·14-s − 1.89e4·16-s + 3.18e4·17-s − 3.64e4·19-s + 1.19e4·20-s − 7.69e4·22-s − 3.83e4·23-s + 1.50e5·25-s + 1.35e5·26-s + 3.35e4·28-s + 1.38e5·29-s + 2.56e5·31-s − 7.13e4·32-s + 3.93e5·34-s + 6.42e5·35-s + 2.21e4·37-s − 4.51e5·38-s − 6.09e5·40-s − 3.16e5·41-s + ⋯
L(s)  = 1  + 1.09·2-s + 0.195·4-s + 1.71·5-s + 1.48·7-s − 0.879·8-s + 1.86·10-s − 1.40·11-s + 1.37·13-s + 1.61·14-s − 1.15·16-s + 1.57·17-s − 1.22·19-s + 0.333·20-s − 1.54·22-s − 0.656·23-s + 1.92·25-s + 1.50·26-s + 0.288·28-s + 1.05·29-s + 1.54·31-s − 0.384·32-s + 1.71·34-s + 2.53·35-s + 0.0717·37-s − 1.33·38-s − 1.50·40-s − 0.716·41-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(6.712301984\)
\(L(\frac12)\) \(\approx\) \(6.712301984\)
\(L(\frac{9}{2})\) 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 \( 1 - 12.3T + 128T^{2} \)
5 \( 1 - 477.T + 7.81e4T^{2} \)
7 \( 1 - 1.34e3T + 8.23e5T^{2} \)
11 \( 1 + 6.22e3T + 1.94e7T^{2} \)
13 \( 1 - 1.09e4T + 6.27e7T^{2} \)
17 \( 1 - 3.18e4T + 4.10e8T^{2} \)
19 \( 1 + 3.64e4T + 8.93e8T^{2} \)
23 \( 1 + 3.83e4T + 3.40e9T^{2} \)
29 \( 1 - 1.38e5T + 1.72e10T^{2} \)
31 \( 1 - 2.56e5T + 2.75e10T^{2} \)
37 \( 1 - 2.21e4T + 9.49e10T^{2} \)
41 \( 1 + 3.16e5T + 1.94e11T^{2} \)
43 \( 1 - 1.48e5T + 2.71e11T^{2} \)
47 \( 1 + 7.48e5T + 5.06e11T^{2} \)
53 \( 1 - 1.94e6T + 1.17e12T^{2} \)
61 \( 1 - 1.76e6T + 3.14e12T^{2} \)
67 \( 1 + 4.25e5T + 6.06e12T^{2} \)
71 \( 1 - 3.19e6T + 9.09e12T^{2} \)
73 \( 1 + 4.35e6T + 1.10e13T^{2} \)
79 \( 1 + 2.91e6T + 1.92e13T^{2} \)
83 \( 1 - 1.97e6T + 2.71e13T^{2} \)
89 \( 1 + 7.72e6T + 4.42e13T^{2} \)
97 \( 1 - 9.26e6T + 8.07e13T^{2} \)
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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

−10.01332526460781253474156774345, −8.580315252438948139014732983453, −8.172035900522729625601196952651, −6.49622300858506817407636267839, −5.68469535762410305449344383843, −5.23203083409136729605132948690, −4.31917188056163373683048400638, −2.92582791602131049420097553435, −2.04223831263835554363170573842, −1.01404936853546914651086428666, 1.01404936853546914651086428666, 2.04223831263835554363170573842, 2.92582791602131049420097553435, 4.31917188056163373683048400638, 5.23203083409136729605132948690, 5.68469535762410305449344383843, 6.49622300858506817407636267839, 8.172035900522729625601196952651, 8.580315252438948139014732983453, 10.01332526460781253474156774345

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