Properties

Label 2-17e2-1.1-c9-0-164
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  − 33.0·2-s + 199.·3-s + 580.·4-s − 108.·5-s − 6.60e3·6-s + 1.47e3·7-s − 2.26e3·8-s + 2.02e4·9-s + 3.59e3·10-s + 8.21e4·11-s + 1.15e5·12-s + 2.36e4·13-s − 4.88e4·14-s − 2.17e4·15-s − 2.22e5·16-s − 6.68e5·18-s − 5.39e5·19-s − 6.31e4·20-s + 2.95e5·21-s − 2.71e6·22-s − 9.60e5·23-s − 4.52e5·24-s − 1.94e6·25-s − 7.80e5·26-s + 1.09e5·27-s + 8.58e5·28-s + 6.36e6·29-s + ⋯
L(s)  = 1  − 1.46·2-s + 1.42·3-s + 1.13·4-s − 0.0778·5-s − 2.08·6-s + 0.232·7-s − 0.195·8-s + 1.02·9-s + 0.113·10-s + 1.69·11-s + 1.61·12-s + 0.229·13-s − 0.339·14-s − 0.110·15-s − 0.848·16-s − 1.50·18-s − 0.948·19-s − 0.0882·20-s + 0.331·21-s − 2.47·22-s − 0.715·23-s − 0.278·24-s − 0.993·25-s − 0.335·26-s + 0.0397·27-s + 0.263·28-s + 1.67·29-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 \( 1 + 33.0T + 512T^{2} \)
3 \( 1 - 199.T + 1.96e4T^{2} \)
5 \( 1 + 108.T + 1.95e6T^{2} \)
7 \( 1 - 1.47e3T + 4.03e7T^{2} \)
11 \( 1 - 8.21e4T + 2.35e9T^{2} \)
13 \( 1 - 2.36e4T + 1.06e10T^{2} \)
19 \( 1 + 5.39e5T + 3.22e11T^{2} \)
23 \( 1 + 9.60e5T + 1.80e12T^{2} \)
29 \( 1 - 6.36e6T + 1.45e13T^{2} \)
31 \( 1 + 5.33e6T + 2.64e13T^{2} \)
37 \( 1 + 2.03e7T + 1.29e14T^{2} \)
41 \( 1 + 3.70e6T + 3.27e14T^{2} \)
43 \( 1 + 3.16e7T + 5.02e14T^{2} \)
47 \( 1 - 3.24e7T + 1.11e15T^{2} \)
53 \( 1 + 3.44e7T + 3.29e15T^{2} \)
59 \( 1 + 8.66e7T + 8.66e15T^{2} \)
61 \( 1 + 2.76e7T + 1.16e16T^{2} \)
67 \( 1 + 8.41e7T + 2.72e16T^{2} \)
71 \( 1 + 1.01e8T + 4.58e16T^{2} \)
73 \( 1 - 3.67e8T + 5.88e16T^{2} \)
79 \( 1 + 1.40e8T + 1.19e17T^{2} \)
83 \( 1 + 7.40e8T + 1.86e17T^{2} \)
89 \( 1 + 2.04e8T + 3.50e17T^{2} \)
97 \( 1 - 7.98e8T + 7.60e17T^{2} \)
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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

−9.481598823424909785137398997104, −8.668361182016970622534630909505, −8.343250697933275673563320703134, −7.28296557236469556808433395014, −6.37690009890503352372390419999, −4.36003841005895920585794525346, −3.41081655679776704990815691744, −1.98458161330464919432981283942, −1.45703332272815662252287485481, 0, 1.45703332272815662252287485481, 1.98458161330464919432981283942, 3.41081655679776704990815691744, 4.36003841005895920585794525346, 6.37690009890503352372390419999, 7.28296557236469556808433395014, 8.343250697933275673563320703134, 8.668361182016970622534630909505, 9.481598823424909785137398997104

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