Properties

Label 2-17e2-1.1-c9-0-45
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  + 28.5·2-s − 173.·3-s + 300.·4-s + 1.46e3·5-s − 4.93e3·6-s − 2.70e3·7-s − 6.03e3·8-s + 1.03e4·9-s + 4.16e4·10-s − 4.51e4·11-s − 5.20e4·12-s − 5.78e4·13-s − 7.71e4·14-s − 2.53e5·15-s − 3.25e5·16-s + 2.94e5·18-s + 3.23e5·19-s + 4.39e5·20-s + 4.68e5·21-s − 1.28e6·22-s − 2.17e6·23-s + 1.04e6·24-s + 1.86e5·25-s − 1.65e6·26-s + 1.62e6·27-s − 8.12e5·28-s + 2.98e6·29-s + ⋯
L(s)  = 1  + 1.25·2-s − 1.23·3-s + 0.586·4-s + 1.04·5-s − 1.55·6-s − 0.425·7-s − 0.520·8-s + 0.524·9-s + 1.31·10-s − 0.930·11-s − 0.724·12-s − 0.562·13-s − 0.536·14-s − 1.29·15-s − 1.24·16-s + 0.660·18-s + 0.570·19-s + 0.614·20-s + 0.525·21-s − 1.17·22-s − 1.62·23-s + 0.642·24-s + 0.0956·25-s − 0.708·26-s + 0.587·27-s − 0.249·28-s + 0.783·29-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

\(L(5)\) \(\approx\) \(1.749736226\)
\(L(\frac12)\) \(\approx\) \(1.749736226\)
\(L(\frac{11}{2})\) 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 \( 1 - 28.5T + 512T^{2} \)
3 \( 1 + 173.T + 1.96e4T^{2} \)
5 \( 1 - 1.46e3T + 1.95e6T^{2} \)
7 \( 1 + 2.70e3T + 4.03e7T^{2} \)
11 \( 1 + 4.51e4T + 2.35e9T^{2} \)
13 \( 1 + 5.78e4T + 1.06e10T^{2} \)
19 \( 1 - 3.23e5T + 3.22e11T^{2} \)
23 \( 1 + 2.17e6T + 1.80e12T^{2} \)
29 \( 1 - 2.98e6T + 1.45e13T^{2} \)
31 \( 1 - 9.05e6T + 2.64e13T^{2} \)
37 \( 1 - 1.28e7T + 1.29e14T^{2} \)
41 \( 1 + 2.91e7T + 3.27e14T^{2} \)
43 \( 1 + 2.54e7T + 5.02e14T^{2} \)
47 \( 1 + 2.66e7T + 1.11e15T^{2} \)
53 \( 1 + 1.32e7T + 3.29e15T^{2} \)
59 \( 1 - 5.04e7T + 8.66e15T^{2} \)
61 \( 1 - 1.10e8T + 1.16e16T^{2} \)
67 \( 1 + 1.13e8T + 2.72e16T^{2} \)
71 \( 1 - 3.38e8T + 4.58e16T^{2} \)
73 \( 1 + 2.36e8T + 5.88e16T^{2} \)
79 \( 1 - 5.61e8T + 1.19e17T^{2} \)
83 \( 1 - 2.95e8T + 1.86e17T^{2} \)
89 \( 1 + 1.78e8T + 3.50e17T^{2} \)
97 \( 1 - 9.76e7T + 7.60e17T^{2} \)
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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

−10.17188275989104237245810200838, −9.822455417166884339064048644725, −8.200485736460265049325051304006, −6.57185028247457666557613331180, −6.10599896418975791743927556242, −5.24789977056349132361324712240, −4.68290845551813631513357124131, −3.17396912561763064279487831137, −2.15119237079576888953535187381, −0.49014341962645575189135711977, 0.49014341962645575189135711977, 2.15119237079576888953535187381, 3.17396912561763064279487831137, 4.68290845551813631513357124131, 5.24789977056349132361324712240, 6.10599896418975791743927556242, 6.57185028247457666557613331180, 8.200485736460265049325051304006, 9.822455417166884339064048644725, 10.17188275989104237245810200838

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