Properties

Label 2-531-1.1-c7-0-137
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  + 22.3·2-s + 370.·4-s + 84.4·5-s + 1.53e3·7-s + 5.40e3·8-s + 1.88e3·10-s − 4.63e3·11-s − 1.71e3·13-s + 3.41e4·14-s + 7.32e4·16-s + 1.42e4·17-s − 2.19e4·19-s + 3.12e4·20-s − 1.03e5·22-s + 1.07e5·23-s − 7.09e4·25-s − 3.81e4·26-s + 5.67e5·28-s + 1.95e5·29-s − 6.81e4·31-s + 9.42e5·32-s + 3.17e5·34-s + 1.29e5·35-s − 3.09e5·37-s − 4.88e5·38-s + 4.56e5·40-s − 8.73e5·41-s + ⋯
L(s)  = 1  + 1.97·2-s + 2.89·4-s + 0.302·5-s + 1.68·7-s + 3.73·8-s + 0.595·10-s − 1.05·11-s − 0.215·13-s + 3.33·14-s + 4.46·16-s + 0.702·17-s − 0.732·19-s + 0.873·20-s − 2.07·22-s + 1.84·23-s − 0.908·25-s − 0.425·26-s + 4.88·28-s + 1.48·29-s − 0.410·31-s + 5.08·32-s + 1.38·34-s + 0.509·35-s − 1.00·37-s − 1.44·38-s + 1.12·40-s − 1.97·41-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

\(L(4)\) \(\approx\) \(12.79010836\)
\(L(\frac12)\) \(\approx\) \(12.79010836\)
\(L(\frac{9}{2})\) 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 \( 1 - 22.3T + 128T^{2} \)
5 \( 1 - 84.4T + 7.81e4T^{2} \)
7 \( 1 - 1.53e3T + 8.23e5T^{2} \)
11 \( 1 + 4.63e3T + 1.94e7T^{2} \)
13 \( 1 + 1.71e3T + 6.27e7T^{2} \)
17 \( 1 - 1.42e4T + 4.10e8T^{2} \)
19 \( 1 + 2.19e4T + 8.93e8T^{2} \)
23 \( 1 - 1.07e5T + 3.40e9T^{2} \)
29 \( 1 - 1.95e5T + 1.72e10T^{2} \)
31 \( 1 + 6.81e4T + 2.75e10T^{2} \)
37 \( 1 + 3.09e5T + 9.49e10T^{2} \)
41 \( 1 + 8.73e5T + 1.94e11T^{2} \)
43 \( 1 - 1.59e5T + 2.71e11T^{2} \)
47 \( 1 - 6.15e5T + 5.06e11T^{2} \)
53 \( 1 - 7.45e5T + 1.17e12T^{2} \)
61 \( 1 + 2.69e6T + 3.14e12T^{2} \)
67 \( 1 - 2.11e6T + 6.06e12T^{2} \)
71 \( 1 - 1.64e6T + 9.09e12T^{2} \)
73 \( 1 + 2.91e6T + 1.10e13T^{2} \)
79 \( 1 - 1.71e5T + 1.92e13T^{2} \)
83 \( 1 + 4.94e6T + 2.71e13T^{2} \)
89 \( 1 - 1.83e6T + 4.42e13T^{2} \)
97 \( 1 + 1.09e7T + 8.07e13T^{2} \)
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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

−10.36547016686777952879295687349, −8.422898641226757308397250932963, −7.60406624432314391514581321553, −6.78018465186836290099351510239, −5.48821045023848692880654378382, −5.12053135547414668091161156251, −4.33388296920702219564927090809, −3.08160494271822639999656703101, −2.17537313731208787327472442532, −1.31123977586898618859132542878, 1.31123977586898618859132542878, 2.17537313731208787327472442532, 3.08160494271822639999656703101, 4.33388296920702219564927090809, 5.12053135547414668091161156251, 5.48821045023848692880654378382, 6.78018465186836290099351510239, 7.60406624432314391514581321553, 8.422898641226757308397250932963, 10.36547016686777952879295687349

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