Properties

Label 2-17e2-1.1-c9-0-35
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.02·2-s − 214.·3-s − 447.·4-s − 357.·5-s + 1.72e3·6-s − 1.17e4·7-s + 7.70e3·8-s + 2.65e4·9-s + 2.86e3·10-s + 7.81e4·11-s + 9.62e4·12-s + 3.03e4·13-s + 9.46e4·14-s + 7.67e4·15-s + 1.67e5·16-s − 2.13e5·18-s − 2.11e5·19-s + 1.59e5·20-s + 2.53e6·21-s − 6.26e5·22-s + 2.01e6·23-s − 1.65e6·24-s − 1.82e6·25-s − 2.43e5·26-s − 1.47e6·27-s + 5.27e6·28-s + 4.53e6·29-s + ⋯
L(s)  = 1  − 0.354·2-s − 1.53·3-s − 0.874·4-s − 0.255·5-s + 0.543·6-s − 1.85·7-s + 0.664·8-s + 1.34·9-s + 0.0906·10-s + 1.60·11-s + 1.33·12-s + 0.294·13-s + 0.658·14-s + 0.391·15-s + 0.638·16-s − 0.478·18-s − 0.372·19-s + 0.223·20-s + 2.84·21-s − 0.570·22-s + 1.50·23-s − 1.01·24-s − 0.934·25-s − 0.104·26-s − 0.533·27-s + 1.62·28-s + 1.18·29-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: \(0\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.4155730102\)
\(L(\frac12)\) \(\approx\) \(0.4155730102\)
\(L(\frac{11}{2})\) 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 \( 1 + 8.02T + 512T^{2} \)
3 \( 1 + 214.T + 1.96e4T^{2} \)
5 \( 1 + 357.T + 1.95e6T^{2} \)
7 \( 1 + 1.17e4T + 4.03e7T^{2} \)
11 \( 1 - 7.81e4T + 2.35e9T^{2} \)
13 \( 1 - 3.03e4T + 1.06e10T^{2} \)
19 \( 1 + 2.11e5T + 3.22e11T^{2} \)
23 \( 1 - 2.01e6T + 1.80e12T^{2} \)
29 \( 1 - 4.53e6T + 1.45e13T^{2} \)
31 \( 1 + 1.97e6T + 2.64e13T^{2} \)
37 \( 1 + 1.66e6T + 1.29e14T^{2} \)
41 \( 1 - 1.02e7T + 3.27e14T^{2} \)
43 \( 1 + 3.61e7T + 5.02e14T^{2} \)
47 \( 1 + 3.61e7T + 1.11e15T^{2} \)
53 \( 1 + 3.15e7T + 3.29e15T^{2} \)
59 \( 1 - 1.04e8T + 8.66e15T^{2} \)
61 \( 1 - 1.28e8T + 1.16e16T^{2} \)
67 \( 1 - 2.11e8T + 2.72e16T^{2} \)
71 \( 1 - 1.32e8T + 4.58e16T^{2} \)
73 \( 1 + 4.13e8T + 5.88e16T^{2} \)
79 \( 1 - 1.92e8T + 1.19e17T^{2} \)
83 \( 1 + 6.55e7T + 1.86e17T^{2} \)
89 \( 1 + 3.70e8T + 3.50e17T^{2} \)
97 \( 1 - 2.80e8T + 7.60e17T^{2} \)
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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

−10.07796979260138108554029130750, −9.516009201800966950575848690405, −8.584368624263629276614703467643, −6.86456042898446442551480363497, −6.47788877326281753489576125401, −5.41061841863576552987063546337, −4.24888568013315511584336183339, −3.40833118076070952424978570673, −1.17697735387924208014667554000, −0.39995944091677713748669176443, 0.39995944091677713748669176443, 1.17697735387924208014667554000, 3.40833118076070952424978570673, 4.24888568013315511584336183339, 5.41061841863576552987063546337, 6.47788877326281753489576125401, 6.86456042898446442551480363497, 8.584368624263629276614703467643, 9.516009201800966950575848690405, 10.07796979260138108554029130750

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