Properties

Label 2-17e2-1.1-c9-0-89
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  − 26.9·2-s + 20.4·3-s + 214.·4-s + 1.22e3·5-s − 549.·6-s − 4.89e3·7-s + 8.02e3·8-s − 1.92e4·9-s − 3.29e4·10-s + 9.03e4·11-s + 4.37e3·12-s + 1.14e5·13-s + 1.31e5·14-s + 2.49e4·15-s − 3.25e5·16-s + 5.19e5·18-s + 8.64e5·19-s + 2.61e5·20-s − 9.98e4·21-s − 2.43e6·22-s + 1.60e6·23-s + 1.63e5·24-s − 4.62e5·25-s − 3.07e6·26-s − 7.94e5·27-s − 1.04e6·28-s + 1.73e6·29-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.145·3-s + 0.418·4-s + 0.873·5-s − 0.173·6-s − 0.770·7-s + 0.692·8-s − 0.978·9-s − 1.04·10-s + 1.86·11-s + 0.0608·12-s + 1.10·13-s + 0.917·14-s + 0.127·15-s − 1.24·16-s + 1.16·18-s + 1.52·19-s + 0.365·20-s − 0.112·21-s − 2.21·22-s + 1.19·23-s + 0.100·24-s − 0.236·25-s − 1.32·26-s − 0.287·27-s − 0.322·28-s + 0.456·29-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

\(L(5)\) \(\approx\) \(1.676482598\)
\(L(\frac12)\) \(\approx\) \(1.676482598\)
\(L(\frac{11}{2})\) 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 \( 1 + 26.9T + 512T^{2} \)
3 \( 1 - 20.4T + 1.96e4T^{2} \)
5 \( 1 - 1.22e3T + 1.95e6T^{2} \)
7 \( 1 + 4.89e3T + 4.03e7T^{2} \)
11 \( 1 - 9.03e4T + 2.35e9T^{2} \)
13 \( 1 - 1.14e5T + 1.06e10T^{2} \)
19 \( 1 - 8.64e5T + 3.22e11T^{2} \)
23 \( 1 - 1.60e6T + 1.80e12T^{2} \)
29 \( 1 - 1.73e6T + 1.45e13T^{2} \)
31 \( 1 + 1.24e6T + 2.64e13T^{2} \)
37 \( 1 + 7.62e5T + 1.29e14T^{2} \)
41 \( 1 - 2.60e7T + 3.27e14T^{2} \)
43 \( 1 - 4.19e7T + 5.02e14T^{2} \)
47 \( 1 - 1.74e7T + 1.11e15T^{2} \)
53 \( 1 - 3.19e7T + 3.29e15T^{2} \)
59 \( 1 + 8.87e7T + 8.66e15T^{2} \)
61 \( 1 - 8.13e7T + 1.16e16T^{2} \)
67 \( 1 - 1.53e8T + 2.72e16T^{2} \)
71 \( 1 + 3.08e8T + 4.58e16T^{2} \)
73 \( 1 + 2.12e8T + 5.88e16T^{2} \)
79 \( 1 + 3.93e8T + 1.19e17T^{2} \)
83 \( 1 + 6.17e8T + 1.86e17T^{2} \)
89 \( 1 - 7.46e8T + 3.50e17T^{2} \)
97 \( 1 - 1.98e7T + 7.60e17T^{2} \)
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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

−9.841465625794382056475986766124, −9.086189329318118585013399155916, −8.911483698764058144697914277136, −7.48489848458417665600711013461, −6.44301028131673231982428310200, −5.65791716367588776337700401485, −3.99600342116912582588106853002, −2.84179743047647674653424032543, −1.40856828759385088436498149711, −0.78476066572236959785604536903, 0.78476066572236959785604536903, 1.40856828759385088436498149711, 2.84179743047647674653424032543, 3.99600342116912582588106853002, 5.65791716367588776337700401485, 6.44301028131673231982428310200, 7.48489848458417665600711013461, 8.911483698764058144697914277136, 9.086189329318118585013399155916, 9.841465625794382056475986766124

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