Properties

Label 2-531-1.1-c7-0-115
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  + 21.6·2-s + 340.·4-s + 399.·5-s − 1.78e3·7-s + 4.59e3·8-s + 8.63e3·10-s + 204.·11-s + 7.46e3·13-s − 3.85e4·14-s + 5.58e4·16-s − 2.07e3·17-s + 3.86e4·19-s + 1.35e5·20-s + 4.41e3·22-s + 4.76e4·23-s + 8.11e4·25-s + 1.61e5·26-s − 6.05e5·28-s − 2.23e5·29-s + 2.73e5·31-s + 6.20e5·32-s − 4.48e4·34-s − 7.10e5·35-s − 5.45e5·37-s + 8.35e5·38-s + 1.83e6·40-s + 5.21e5·41-s + ⋯
L(s)  = 1  + 1.91·2-s + 2.65·4-s + 1.42·5-s − 1.96·7-s + 3.17·8-s + 2.73·10-s + 0.0462·11-s + 0.942·13-s − 3.75·14-s + 3.40·16-s − 0.102·17-s + 1.29·19-s + 3.79·20-s + 0.0884·22-s + 0.815·23-s + 1.03·25-s + 1.80·26-s − 5.21·28-s − 1.70·29-s + 1.65·31-s + 3.34·32-s − 0.195·34-s − 2.80·35-s − 1.76·37-s + 2.47·38-s + 4.52·40-s + 1.18·41-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

\(L(4)\) \(\approx\) \(10.78283595\)
\(L(\frac12)\) \(\approx\) \(10.78283595\)
\(L(\frac{9}{2})\) 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 \( 1 - 21.6T + 128T^{2} \)
5 \( 1 - 399.T + 7.81e4T^{2} \)
7 \( 1 + 1.78e3T + 8.23e5T^{2} \)
11 \( 1 - 204.T + 1.94e7T^{2} \)
13 \( 1 - 7.46e3T + 6.27e7T^{2} \)
17 \( 1 + 2.07e3T + 4.10e8T^{2} \)
19 \( 1 - 3.86e4T + 8.93e8T^{2} \)
23 \( 1 - 4.76e4T + 3.40e9T^{2} \)
29 \( 1 + 2.23e5T + 1.72e10T^{2} \)
31 \( 1 - 2.73e5T + 2.75e10T^{2} \)
37 \( 1 + 5.45e5T + 9.49e10T^{2} \)
41 \( 1 - 5.21e5T + 1.94e11T^{2} \)
43 \( 1 - 3.27e5T + 2.71e11T^{2} \)
47 \( 1 - 2.41e4T + 5.06e11T^{2} \)
53 \( 1 - 1.06e6T + 1.17e12T^{2} \)
61 \( 1 - 1.08e6T + 3.14e12T^{2} \)
67 \( 1 + 5.57e4T + 6.06e12T^{2} \)
71 \( 1 - 8.51e5T + 9.09e12T^{2} \)
73 \( 1 - 3.44e6T + 1.10e13T^{2} \)
79 \( 1 + 8.05e6T + 1.92e13T^{2} \)
83 \( 1 + 2.05e6T + 2.71e13T^{2} \)
89 \( 1 - 1.68e6T + 4.42e13T^{2} \)
97 \( 1 - 1.14e7T + 8.07e13T^{2} \)
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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

−9.919001043558634489362967636589, −9.104690042615760026248246853601, −7.21095307015551234461221451844, −6.54364962580712952823425909319, −5.85298858165566476283427289774, −5.35850199641265882214511554434, −3.88635987323666227604642525407, −3.14526195594466263523909600591, −2.38514470028367200083117757569, −1.12772725536828263795942129337, 1.12772725536828263795942129337, 2.38514470028367200083117757569, 3.14526195594466263523909600591, 3.88635987323666227604642525407, 5.35850199641265882214511554434, 5.85298858165566476283427289774, 6.54364962580712952823425909319, 7.21095307015551234461221451844, 9.104690042615760026248246853601, 9.919001043558634489362967636589

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