Properties

Label 2-17e2-1.1-c9-0-8
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  + 5.41·2-s − 128.·3-s − 482.·4-s − 1.44e3·5-s − 696.·6-s + 9.44e3·7-s − 5.38e3·8-s − 3.14e3·9-s − 7.84e3·10-s − 5.33e4·11-s + 6.20e4·12-s + 5.04e4·13-s + 5.11e4·14-s + 1.86e5·15-s + 2.17e5·16-s − 1.70e4·18-s − 3.61e5·19-s + 6.99e5·20-s − 1.21e6·21-s − 2.88e5·22-s − 3.78e5·23-s + 6.92e5·24-s + 1.46e5·25-s + 2.73e5·26-s + 2.93e6·27-s − 4.55e6·28-s − 4.53e6·29-s + ⋯
L(s)  = 1  + 0.239·2-s − 0.916·3-s − 0.942·4-s − 1.03·5-s − 0.219·6-s + 1.48·7-s − 0.464·8-s − 0.159·9-s − 0.248·10-s − 1.09·11-s + 0.864·12-s + 0.490·13-s + 0.355·14-s + 0.950·15-s + 0.831·16-s − 0.0382·18-s − 0.635·19-s + 0.977·20-s − 1.36·21-s − 0.262·22-s − 0.281·23-s + 0.426·24-s + 0.0749·25-s + 0.117·26-s + 1.06·27-s − 1.40·28-s − 1.19·29-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

\(L(5)\) \(\approx\) \(0.1217755545\)
\(L(\frac12)\) \(\approx\) \(0.1217755545\)
\(L(\frac{11}{2})\) 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 \( 1 - 5.41T + 512T^{2} \)
3 \( 1 + 128.T + 1.96e4T^{2} \)
5 \( 1 + 1.44e3T + 1.95e6T^{2} \)
7 \( 1 - 9.44e3T + 4.03e7T^{2} \)
11 \( 1 + 5.33e4T + 2.35e9T^{2} \)
13 \( 1 - 5.04e4T + 1.06e10T^{2} \)
19 \( 1 + 3.61e5T + 3.22e11T^{2} \)
23 \( 1 + 3.78e5T + 1.80e12T^{2} \)
29 \( 1 + 4.53e6T + 1.45e13T^{2} \)
31 \( 1 + 7.59e6T + 2.64e13T^{2} \)
37 \( 1 + 1.33e7T + 1.29e14T^{2} \)
41 \( 1 - 1.88e7T + 3.27e14T^{2} \)
43 \( 1 + 1.12e7T + 5.02e14T^{2} \)
47 \( 1 + 2.82e7T + 1.11e15T^{2} \)
53 \( 1 + 1.10e8T + 3.29e15T^{2} \)
59 \( 1 + 1.66e8T + 8.66e15T^{2} \)
61 \( 1 + 1.24e8T + 1.16e16T^{2} \)
67 \( 1 + 2.73e8T + 2.72e16T^{2} \)
71 \( 1 - 1.28e8T + 4.58e16T^{2} \)
73 \( 1 + 1.38e8T + 5.88e16T^{2} \)
79 \( 1 - 6.51e8T + 1.19e17T^{2} \)
83 \( 1 - 2.90e8T + 1.86e17T^{2} \)
89 \( 1 + 8.64e7T + 3.50e17T^{2} \)
97 \( 1 + 9.69e8T + 7.60e17T^{2} \)
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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

−10.74899131989648730034601848353, −9.114222899322469550759676964998, −8.141481553426137847542966545431, −7.65909881070010406679935164317, −5.95730918770157675535348818915, −5.11733618702326442041169747463, −4.49591992791473200579576062602, −3.40282392597662263011291316264, −1.63284985826375574167989332086, −0.15926175768030139552009914101, 0.15926175768030139552009914101, 1.63284985826375574167989332086, 3.40282392597662263011291316264, 4.49591992791473200579576062602, 5.11733618702326442041169747463, 5.95730918770157675535348818915, 7.65909881070010406679935164317, 8.141481553426137847542966545431, 9.114222899322469550759676964998, 10.74899131989648730034601848353

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