Properties

Label 2-17e2-1.1-c9-0-106
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  + 26.5·2-s − 204.·3-s + 194.·4-s − 2.37e3·5-s − 5.44e3·6-s + 6.24e3·7-s − 8.43e3·8-s + 2.23e4·9-s − 6.30e4·10-s − 8.26e4·11-s − 3.98e4·12-s + 1.10e5·13-s + 1.65e5·14-s + 4.86e5·15-s − 3.23e5·16-s + 5.93e5·18-s + 5.62e5·19-s − 4.61e5·20-s − 1.27e6·21-s − 2.19e6·22-s − 1.19e6·23-s + 1.72e6·24-s + 3.67e6·25-s + 2.94e6·26-s − 5.40e5·27-s + 1.21e6·28-s + 1.98e6·29-s + ⋯
L(s)  = 1  + 1.17·2-s − 1.46·3-s + 0.379·4-s − 1.69·5-s − 1.71·6-s + 0.982·7-s − 0.728·8-s + 1.13·9-s − 1.99·10-s − 1.70·11-s − 0.555·12-s + 1.07·13-s + 1.15·14-s + 2.47·15-s − 1.23·16-s + 1.33·18-s + 0.989·19-s − 0.644·20-s − 1.43·21-s − 1.99·22-s − 0.890·23-s + 1.06·24-s + 1.88·25-s + 1.26·26-s − 0.195·27-s + 0.373·28-s + 0.521·29-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 \( 1 - 26.5T + 512T^{2} \)
3 \( 1 + 204.T + 1.96e4T^{2} \)
5 \( 1 + 2.37e3T + 1.95e6T^{2} \)
7 \( 1 - 6.24e3T + 4.03e7T^{2} \)
11 \( 1 + 8.26e4T + 2.35e9T^{2} \)
13 \( 1 - 1.10e5T + 1.06e10T^{2} \)
19 \( 1 - 5.62e5T + 3.22e11T^{2} \)
23 \( 1 + 1.19e6T + 1.80e12T^{2} \)
29 \( 1 - 1.98e6T + 1.45e13T^{2} \)
31 \( 1 - 9.43e6T + 2.64e13T^{2} \)
37 \( 1 - 9.84e5T + 1.29e14T^{2} \)
41 \( 1 - 2.19e6T + 3.27e14T^{2} \)
43 \( 1 - 3.42e6T + 5.02e14T^{2} \)
47 \( 1 + 2.11e7T + 1.11e15T^{2} \)
53 \( 1 - 2.19e7T + 3.29e15T^{2} \)
59 \( 1 - 4.21e7T + 8.66e15T^{2} \)
61 \( 1 + 6.01e7T + 1.16e16T^{2} \)
67 \( 1 - 2.11e8T + 2.72e16T^{2} \)
71 \( 1 + 3.37e8T + 4.58e16T^{2} \)
73 \( 1 - 2.34e8T + 5.88e16T^{2} \)
79 \( 1 + 1.77e8T + 1.19e17T^{2} \)
83 \( 1 - 1.29e8T + 1.86e17T^{2} \)
89 \( 1 - 2.82e8T + 3.50e17T^{2} \)
97 \( 1 + 4.77e7T + 7.60e17T^{2} \)
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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

−10.33678215641007014960018856379, −8.368868250382678458247861896321, −7.77338192838365187752773642342, −6.48485681150258261892662822650, −5.42514685710725072995948442689, −4.82531224251414892304257716570, −4.08603734036040459208355532506, −2.92336098977872768297774769859, −0.893254292262306518877130073058, 0, 0.893254292262306518877130073058, 2.92336098977872768297774769859, 4.08603734036040459208355532506, 4.82531224251414892304257716570, 5.42514685710725072995948442689, 6.48485681150258261892662822650, 7.77338192838365187752773642342, 8.368868250382678458247861896321, 10.33678215641007014960018856379

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