Properties

Label 2-17e2-1.1-c9-0-115
Degree $2$
Conductor $289$
Sign $-1$
Analytic cond. $148.845$
Root an. cond. $12.2002$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.22·2-s − 177.·3-s − 510.·4-s + 1.62e3·5-s + 217.·6-s + 1.83e3·7-s + 1.25e3·8-s + 1.18e4·9-s − 1.98e3·10-s + 3.17e4·11-s + 9.05e4·12-s − 1.32e5·13-s − 2.25e3·14-s − 2.87e5·15-s + 2.59e5·16-s − 1.44e4·18-s − 1.60e3·19-s − 8.27e5·20-s − 3.25e5·21-s − 3.90e4·22-s − 2.32e4·23-s − 2.22e5·24-s + 6.71e5·25-s + 1.62e5·26-s + 1.39e6·27-s − 9.36e5·28-s − 3.73e6·29-s + ⋯
L(s)  = 1  − 0.0542·2-s − 1.26·3-s − 0.997·4-s + 1.15·5-s + 0.0685·6-s + 0.288·7-s + 0.108·8-s + 0.599·9-s − 0.0628·10-s + 0.654·11-s + 1.26·12-s − 1.28·13-s − 0.0156·14-s − 1.46·15-s + 0.991·16-s − 0.0325·18-s − 0.00282·19-s − 1.15·20-s − 0.365·21-s − 0.0354·22-s − 0.0173·23-s − 0.136·24-s + 0.343·25-s + 0.0697·26-s + 0.506·27-s − 0.287·28-s − 0.980·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-1$
Analytic conductor: \(148.845\)
Root analytic conductor: \(12.2002\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + 1.22T + 512T^{2} \)
3 \( 1 + 177.T + 1.96e4T^{2} \)
5 \( 1 - 1.62e3T + 1.95e6T^{2} \)
7 \( 1 - 1.83e3T + 4.03e7T^{2} \)
11 \( 1 - 3.17e4T + 2.35e9T^{2} \)
13 \( 1 + 1.32e5T + 1.06e10T^{2} \)
19 \( 1 + 1.60e3T + 3.22e11T^{2} \)
23 \( 1 + 2.32e4T + 1.80e12T^{2} \)
29 \( 1 + 3.73e6T + 1.45e13T^{2} \)
31 \( 1 + 8.91e6T + 2.64e13T^{2} \)
37 \( 1 - 1.20e7T + 1.29e14T^{2} \)
41 \( 1 - 1.26e7T + 3.27e14T^{2} \)
43 \( 1 - 2.86e7T + 5.02e14T^{2} \)
47 \( 1 + 7.17e6T + 1.11e15T^{2} \)
53 \( 1 + 5.96e7T + 3.29e15T^{2} \)
59 \( 1 - 1.85e8T + 8.66e15T^{2} \)
61 \( 1 - 2.00e8T + 1.16e16T^{2} \)
67 \( 1 + 1.27e8T + 2.72e16T^{2} \)
71 \( 1 - 3.27e8T + 4.58e16T^{2} \)
73 \( 1 - 1.48e8T + 5.88e16T^{2} \)
79 \( 1 - 2.58e8T + 1.19e17T^{2} \)
83 \( 1 - 3.45e8T + 1.86e17T^{2} \)
89 \( 1 - 4.03e8T + 3.50e17T^{2} \)
97 \( 1 - 9.89e8T + 7.60e17T^{2} \)
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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

−9.629590164094853740132163455110, −9.255038550277111887503946204132, −7.78096681876885983556066987776, −6.56533502824756570202659440174, −5.54680704763194872130616140322, −5.12992087809474113921163579496, −3.97626057774370387864175193400, −2.19495172475593147210424924159, −1.00021142971809066942000792203, 0, 1.00021142971809066942000792203, 2.19495172475593147210424924159, 3.97626057774370387864175193400, 5.12992087809474113921163579496, 5.54680704763194872130616140322, 6.56533502824756570202659440174, 7.78096681876885983556066987776, 9.255038550277111887503946204132, 9.629590164094853740132163455110

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