Properties

Label 2-17e2-1.1-c9-0-191
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 31.0·2-s + 53.8·3-s + 449.·4-s + 1.14e3·5-s + 1.66e3·6-s + 1.00e4·7-s − 1.93e3·8-s − 1.67e4·9-s + 3.54e4·10-s − 1.92e4·11-s + 2.41e4·12-s − 1.49e5·13-s + 3.11e5·14-s + 6.14e4·15-s − 2.90e5·16-s − 5.20e5·18-s − 7.89e5·19-s + 5.13e5·20-s + 5.40e5·21-s − 5.98e5·22-s + 6.49e5·23-s − 1.04e5·24-s − 6.49e5·25-s − 4.63e6·26-s − 1.96e6·27-s + 4.51e6·28-s − 5.71e6·29-s + ⋯
L(s)  = 1  + 1.37·2-s + 0.383·3-s + 0.877·4-s + 0.816·5-s + 0.525·6-s + 1.58·7-s − 0.167·8-s − 0.852·9-s + 1.11·10-s − 0.397·11-s + 0.336·12-s − 1.45·13-s + 2.16·14-s + 0.313·15-s − 1.10·16-s − 1.16·18-s − 1.38·19-s + 0.717·20-s + 0.606·21-s − 0.544·22-s + 0.484·23-s − 0.0642·24-s − 0.332·25-s − 1.98·26-s − 0.710·27-s + 1.38·28-s − 1.50·29-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: \(1\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 31.0T + 512T^{2} \)
3 \( 1 - 53.8T + 1.96e4T^{2} \)
5 \( 1 - 1.14e3T + 1.95e6T^{2} \)
7 \( 1 - 1.00e4T + 4.03e7T^{2} \)
11 \( 1 + 1.92e4T + 2.35e9T^{2} \)
13 \( 1 + 1.49e5T + 1.06e10T^{2} \)
19 \( 1 + 7.89e5T + 3.22e11T^{2} \)
23 \( 1 - 6.49e5T + 1.80e12T^{2} \)
29 \( 1 + 5.71e6T + 1.45e13T^{2} \)
31 \( 1 - 1.39e5T + 2.64e13T^{2} \)
37 \( 1 + 1.07e5T + 1.29e14T^{2} \)
41 \( 1 - 2.16e6T + 3.27e14T^{2} \)
43 \( 1 + 3.93e7T + 5.02e14T^{2} \)
47 \( 1 + 2.88e7T + 1.11e15T^{2} \)
53 \( 1 + 1.39e7T + 3.29e15T^{2} \)
59 \( 1 - 7.21e7T + 8.66e15T^{2} \)
61 \( 1 - 1.66e8T + 1.16e16T^{2} \)
67 \( 1 - 2.47e8T + 2.72e16T^{2} \)
71 \( 1 - 2.68e8T + 4.58e16T^{2} \)
73 \( 1 - 3.16e7T + 5.88e16T^{2} \)
79 \( 1 + 4.91e7T + 1.19e17T^{2} \)
83 \( 1 - 8.25e8T + 1.86e17T^{2} \)
89 \( 1 + 1.30e8T + 3.50e17T^{2} \)
97 \( 1 - 8.84e8T + 7.60e17T^{2} \)
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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

−9.786526594718326131696391033738, −8.703444583297734715794082537843, −7.81215241913791618175197128683, −6.51947899964887728170126643638, −5.25987628826348012304515537590, −5.07288021638410762151964172426, −3.78671064068708694286979985035, −2.39042137567555050359972184087, −2.01773244699953122576685625861, 0, 2.01773244699953122576685625861, 2.39042137567555050359972184087, 3.78671064068708694286979985035, 5.07288021638410762151964172426, 5.25987628826348012304515537590, 6.51947899964887728170126643638, 7.81215241913791618175197128683, 8.703444583297734715794082537843, 9.786526594718326131696391033738

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