Properties

Label 2-17e2-1.1-c9-0-187
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  + 35.6·2-s − 0.238·3-s + 757.·4-s + 2.00e3·5-s − 8.48·6-s − 3.11e3·7-s + 8.74e3·8-s − 1.96e4·9-s + 7.14e4·10-s + 2.19e4·11-s − 180.·12-s − 1.70e5·13-s − 1.10e5·14-s − 477.·15-s − 7.62e4·16-s − 7.01e5·18-s + 5.78e5·19-s + 1.51e6·20-s + 740.·21-s + 7.83e5·22-s − 1.14e6·23-s − 2.08e3·24-s + 2.07e6·25-s − 6.06e6·26-s + 9.37e3·27-s − 2.35e6·28-s − 2.04e6·29-s + ⋯
L(s)  = 1  + 1.57·2-s − 0.00169·3-s + 1.47·4-s + 1.43·5-s − 0.00267·6-s − 0.489·7-s + 0.754·8-s − 0.999·9-s + 2.26·10-s + 0.452·11-s − 0.00251·12-s − 1.65·13-s − 0.771·14-s − 0.00243·15-s − 0.290·16-s − 1.57·18-s + 1.01·19-s + 2.12·20-s + 0.000830·21-s + 0.713·22-s − 0.850·23-s − 0.00128·24-s + 1.06·25-s − 2.60·26-s + 0.00339·27-s − 0.724·28-s − 0.536·29-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 \( 1 - 35.6T + 512T^{2} \)
3 \( 1 + 0.238T + 1.96e4T^{2} \)
5 \( 1 - 2.00e3T + 1.95e6T^{2} \)
7 \( 1 + 3.11e3T + 4.03e7T^{2} \)
11 \( 1 - 2.19e4T + 2.35e9T^{2} \)
13 \( 1 + 1.70e5T + 1.06e10T^{2} \)
19 \( 1 - 5.78e5T + 3.22e11T^{2} \)
23 \( 1 + 1.14e6T + 1.80e12T^{2} \)
29 \( 1 + 2.04e6T + 1.45e13T^{2} \)
31 \( 1 + 6.36e6T + 2.64e13T^{2} \)
37 \( 1 + 1.74e7T + 1.29e14T^{2} \)
41 \( 1 + 8.92e6T + 3.27e14T^{2} \)
43 \( 1 - 2.23e7T + 5.02e14T^{2} \)
47 \( 1 - 5.29e7T + 1.11e15T^{2} \)
53 \( 1 + 4.96e7T + 3.29e15T^{2} \)
59 \( 1 + 1.25e8T + 8.66e15T^{2} \)
61 \( 1 - 7.55e7T + 1.16e16T^{2} \)
67 \( 1 + 5.66e7T + 2.72e16T^{2} \)
71 \( 1 - 9.81e6T + 4.58e16T^{2} \)
73 \( 1 - 4.37e8T + 5.88e16T^{2} \)
79 \( 1 + 1.18e8T + 1.19e17T^{2} \)
83 \( 1 - 7.42e8T + 1.86e17T^{2} \)
89 \( 1 + 6.40e8T + 3.50e17T^{2} \)
97 \( 1 + 1.49e9T + 7.60e17T^{2} \)
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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

−9.721640089348038163670247971756, −9.134671895866169471958467480866, −7.37431617016293442819913992729, −6.36934519194449430887387941423, −5.58224657738572262573092219640, −5.05957102328826537611737147979, −3.61303672871004094428310747501, −2.65585966640354173850711015313, −1.89595291920455956341818124363, 0, 1.89595291920455956341818124363, 2.65585966640354173850711015313, 3.61303672871004094428310747501, 5.05957102328826537611737147979, 5.58224657738572262573092219640, 6.36934519194449430887387941423, 7.37431617016293442819913992729, 9.134671895866169471958467480866, 9.721640089348038163670247971756

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