Properties

Label 2-17e2-1.1-c9-0-121
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  − 37.5·2-s − 70.9·3-s + 898.·4-s + 98.5·5-s + 2.66e3·6-s + 4.15e3·7-s − 1.44e4·8-s − 1.46e4·9-s − 3.70e3·10-s + 1.76e4·11-s − 6.37e4·12-s + 1.25e5·13-s − 1.56e5·14-s − 6.99e3·15-s + 8.46e4·16-s + 5.50e5·18-s + 1.86e5·19-s + 8.84e4·20-s − 2.95e5·21-s − 6.63e5·22-s + 8.80e5·23-s + 1.02e6·24-s − 1.94e6·25-s − 4.70e6·26-s + 2.43e6·27-s + 3.73e6·28-s + 4.72e5·29-s + ⋯
L(s)  = 1  − 1.65·2-s − 0.505·3-s + 1.75·4-s + 0.0705·5-s + 0.839·6-s + 0.654·7-s − 1.25·8-s − 0.744·9-s − 0.117·10-s + 0.363·11-s − 0.887·12-s + 1.21·13-s − 1.08·14-s − 0.0356·15-s + 0.322·16-s + 1.23·18-s + 0.328·19-s + 0.123·20-s − 0.331·21-s − 0.603·22-s + 0.656·23-s + 0.632·24-s − 0.995·25-s − 2.01·26-s + 0.882·27-s + 1.14·28-s + 0.124·29-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 \( 1 + 37.5T + 512T^{2} \)
3 \( 1 + 70.9T + 1.96e4T^{2} \)
5 \( 1 - 98.5T + 1.95e6T^{2} \)
7 \( 1 - 4.15e3T + 4.03e7T^{2} \)
11 \( 1 - 1.76e4T + 2.35e9T^{2} \)
13 \( 1 - 1.25e5T + 1.06e10T^{2} \)
19 \( 1 - 1.86e5T + 3.22e11T^{2} \)
23 \( 1 - 8.80e5T + 1.80e12T^{2} \)
29 \( 1 - 4.72e5T + 1.45e13T^{2} \)
31 \( 1 - 5.47e6T + 2.64e13T^{2} \)
37 \( 1 + 1.99e7T + 1.29e14T^{2} \)
41 \( 1 + 1.70e7T + 3.27e14T^{2} \)
43 \( 1 + 2.41e7T + 5.02e14T^{2} \)
47 \( 1 + 3.99e7T + 1.11e15T^{2} \)
53 \( 1 + 3.20e7T + 3.29e15T^{2} \)
59 \( 1 - 1.43e8T + 8.66e15T^{2} \)
61 \( 1 - 9.63e7T + 1.16e16T^{2} \)
67 \( 1 - 1.54e8T + 2.72e16T^{2} \)
71 \( 1 + 3.43e8T + 4.58e16T^{2} \)
73 \( 1 + 1.81e8T + 5.88e16T^{2} \)
79 \( 1 - 4.31e8T + 1.19e17T^{2} \)
83 \( 1 + 3.37e8T + 1.86e17T^{2} \)
89 \( 1 + 2.13e8T + 3.50e17T^{2} \)
97 \( 1 - 5.74e8T + 7.60e17T^{2} \)
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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

−9.755313563742151597466913396703, −8.518585458418115700925614420244, −8.368895863747872266320494789846, −7.00732987821419793583455556329, −6.17065072112438345177498089573, −4.99797318028314090041067269308, −3.32899599301678660415856073679, −1.86241045795530318150299825127, −1.04766819940080150066404076693, 0, 1.04766819940080150066404076693, 1.86241045795530318150299825127, 3.32899599301678660415856073679, 4.99797318028314090041067269308, 6.17065072112438345177498089573, 7.00732987821419793583455556329, 8.368895863747872266320494789846, 8.518585458418115700925614420244, 9.755313563742151597466913396703

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