Properties

Label 2-17e2-1.1-c9-0-50
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  − 24.0·2-s − 194.·3-s + 64.3·4-s + 2.54e3·5-s + 4.66e3·6-s − 1.96e3·7-s + 1.07e4·8-s + 1.80e4·9-s − 6.10e4·10-s + 1.82e4·11-s − 1.24e4·12-s − 1.65e5·13-s + 4.71e4·14-s − 4.94e5·15-s − 2.90e5·16-s − 4.33e5·18-s + 1.37e5·19-s + 1.63e5·20-s + 3.81e5·21-s − 4.37e5·22-s + 6.95e5·23-s − 2.08e6·24-s + 4.51e6·25-s + 3.97e6·26-s + 3.14e5·27-s − 1.26e5·28-s + 4.23e6·29-s + ⋯
L(s)  = 1  − 1.06·2-s − 1.38·3-s + 0.125·4-s + 1.82·5-s + 1.46·6-s − 0.308·7-s + 0.927·8-s + 0.917·9-s − 1.93·10-s + 0.375·11-s − 0.173·12-s − 1.60·13-s + 0.327·14-s − 2.52·15-s − 1.10·16-s − 0.973·18-s + 0.242·19-s + 0.228·20-s + 0.427·21-s − 0.398·22-s + 0.518·23-s − 1.28·24-s + 2.31·25-s + 1.70·26-s + 0.113·27-s − 0.0388·28-s + 1.11·29-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

\(L(5)\) \(\approx\) \(0.7994872745\)
\(L(\frac12)\) \(\approx\) \(0.7994872745\)
\(L(\frac{11}{2})\) 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 \( 1 + 24.0T + 512T^{2} \)
3 \( 1 + 194.T + 1.96e4T^{2} \)
5 \( 1 - 2.54e3T + 1.95e6T^{2} \)
7 \( 1 + 1.96e3T + 4.03e7T^{2} \)
11 \( 1 - 1.82e4T + 2.35e9T^{2} \)
13 \( 1 + 1.65e5T + 1.06e10T^{2} \)
19 \( 1 - 1.37e5T + 3.22e11T^{2} \)
23 \( 1 - 6.95e5T + 1.80e12T^{2} \)
29 \( 1 - 4.23e6T + 1.45e13T^{2} \)
31 \( 1 - 3.78e6T + 2.64e13T^{2} \)
37 \( 1 + 1.18e7T + 1.29e14T^{2} \)
41 \( 1 - 9.83e6T + 3.27e14T^{2} \)
43 \( 1 + 3.54e7T + 5.02e14T^{2} \)
47 \( 1 - 5.08e7T + 1.11e15T^{2} \)
53 \( 1 - 1.55e7T + 3.29e15T^{2} \)
59 \( 1 + 9.29e7T + 8.66e15T^{2} \)
61 \( 1 - 2.04e8T + 1.16e16T^{2} \)
67 \( 1 - 5.35e7T + 2.72e16T^{2} \)
71 \( 1 - 1.33e8T + 4.58e16T^{2} \)
73 \( 1 + 4.37e7T + 5.88e16T^{2} \)
79 \( 1 + 1.37e8T + 1.19e17T^{2} \)
83 \( 1 + 4.10e8T + 1.86e17T^{2} \)
89 \( 1 - 2.89e8T + 3.50e17T^{2} \)
97 \( 1 - 2.82e8T + 7.60e17T^{2} \)
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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

−10.02989562496337023939331070472, −9.660667490528133851955037247042, −8.649179640258942869189879114273, −7.08496047962879544530749417218, −6.45222101890620812557566317008, −5.33859024125534932798882262964, −4.77453011164786959089527620837, −2.54392085619097919128486027719, −1.41929247745945625452661093093, −0.54688448334261508572496782600, 0.54688448334261508572496782600, 1.41929247745945625452661093093, 2.54392085619097919128486027719, 4.77453011164786959089527620837, 5.33859024125534932798882262964, 6.45222101890620812557566317008, 7.08496047962879544530749417218, 8.649179640258942869189879114273, 9.660667490528133851955037247042, 10.02989562496337023939331070472

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