Properties

Label 2-17e2-1.1-c9-0-178
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  + 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: \(1\)
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 \( 1 - 28.5T + 512T^{2} \)
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

−9.382297278286639636034316682271, −8.847293115405812317137648387172, −7.72449131730079775940894166584, −6.97023807595512390038709604847, −5.46781530645413017511977786494, −4.45631879603073114592118782912, −3.59557088903626831312911921741, −3.03867319834750320736083996382, −1.70237398028773029207320709307, 0, 1.70237398028773029207320709307, 3.03867319834750320736083996382, 3.59557088903626831312911921741, 4.45631879603073114592118782912, 5.46781530645413017511977786494, 6.97023807595512390038709604847, 7.72449131730079775940894166584, 8.847293115405812317137648387172, 9.382297278286639636034316682271

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