Properties

Label 2-17e2-1.1-c9-0-156
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  − 25.8·2-s + 225.·3-s + 154.·4-s + 96.4·5-s − 5.81e3·6-s − 1.38e3·7-s + 9.22e3·8-s + 3.10e4·9-s − 2.49e3·10-s − 3.38e4·11-s + 3.48e4·12-s + 1.38e5·13-s + 3.57e4·14-s + 2.17e4·15-s − 3.17e5·16-s − 8.00e5·18-s − 4.29e5·19-s + 1.49e4·20-s − 3.12e5·21-s + 8.73e5·22-s + 3.52e5·23-s + 2.07e6·24-s − 1.94e6·25-s − 3.57e6·26-s + 2.54e6·27-s − 2.14e5·28-s − 5.08e5·29-s + ⋯
L(s)  = 1  − 1.14·2-s + 1.60·3-s + 0.302·4-s + 0.0690·5-s − 1.83·6-s − 0.218·7-s + 0.796·8-s + 1.57·9-s − 0.0787·10-s − 0.696·11-s + 0.485·12-s + 1.34·13-s + 0.248·14-s + 0.110·15-s − 1.21·16-s − 1.79·18-s − 0.756·19-s + 0.0208·20-s − 0.350·21-s + 0.795·22-s + 0.262·23-s + 1.27·24-s − 0.995·25-s − 1.53·26-s + 0.923·27-s − 0.0659·28-s − 0.133·29-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: Trivial
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 \( 1 + 25.8T + 512T^{2} \)
3 \( 1 - 225.T + 1.96e4T^{2} \)
5 \( 1 - 96.4T + 1.95e6T^{2} \)
7 \( 1 + 1.38e3T + 4.03e7T^{2} \)
11 \( 1 + 3.38e4T + 2.35e9T^{2} \)
13 \( 1 - 1.38e5T + 1.06e10T^{2} \)
19 \( 1 + 4.29e5T + 3.22e11T^{2} \)
23 \( 1 - 3.52e5T + 1.80e12T^{2} \)
29 \( 1 + 5.08e5T + 1.45e13T^{2} \)
31 \( 1 + 8.73e6T + 2.64e13T^{2} \)
37 \( 1 - 1.62e7T + 1.29e14T^{2} \)
41 \( 1 - 1.12e7T + 3.27e14T^{2} \)
43 \( 1 - 2.43e7T + 5.02e14T^{2} \)
47 \( 1 + 1.11e7T + 1.11e15T^{2} \)
53 \( 1 + 3.54e7T + 3.29e15T^{2} \)
59 \( 1 + 5.03e7T + 8.66e15T^{2} \)
61 \( 1 + 1.53e8T + 1.16e16T^{2} \)
67 \( 1 + 2.44e8T + 2.72e16T^{2} \)
71 \( 1 + 3.78e8T + 4.58e16T^{2} \)
73 \( 1 - 1.38e7T + 5.88e16T^{2} \)
79 \( 1 - 5.65e8T + 1.19e17T^{2} \)
83 \( 1 - 5.68e8T + 1.86e17T^{2} \)
89 \( 1 - 2.61e8T + 3.50e17T^{2} \)
97 \( 1 + 1.04e9T + 7.60e17T^{2} \)
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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

−9.275180302960141281466122360639, −9.008012101516819221438313190517, −7.944338047752003042769381822862, −7.59107678632131907759435973064, −6.10852316189155076031288152411, −4.38568854561951730435667927614, −3.40848825183125641502571834361, −2.23646306710197895425660814904, −1.37015951443949435443936759767, 0, 1.37015951443949435443936759767, 2.23646306710197895425660814904, 3.40848825183125641502571834361, 4.38568854561951730435667927614, 6.10852316189155076031288152411, 7.59107678632131907759435973064, 7.944338047752003042769381822862, 9.008012101516819221438313190517, 9.275180302960141281466122360639

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