Properties

Label 2-17e2-1.1-c9-0-82
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  − 44.7·2-s − 202.·3-s + 1.48e3·4-s − 1.68e3·5-s + 9.06e3·6-s + 4.58e3·7-s − 4.37e4·8-s + 2.13e4·9-s + 7.54e4·10-s − 2.91e4·11-s − 3.01e5·12-s + 5.96e4·13-s − 2.05e5·14-s + 3.41e5·15-s + 1.19e6·16-s − 9.54e5·18-s + 3.93e5·19-s − 2.51e6·20-s − 9.29e5·21-s + 1.30e6·22-s + 5.81e5·23-s + 8.85e6·24-s + 8.87e5·25-s − 2.66e6·26-s − 3.33e5·27-s + 6.83e6·28-s − 5.69e6·29-s + ⋯
L(s)  = 1  − 1.97·2-s − 1.44·3-s + 2.90·4-s − 1.20·5-s + 2.85·6-s + 0.722·7-s − 3.77·8-s + 1.08·9-s + 2.38·10-s − 0.600·11-s − 4.19·12-s + 0.579·13-s − 1.42·14-s + 1.74·15-s + 4.55·16-s − 2.14·18-s + 0.692·19-s − 3.50·20-s − 1.04·21-s + 1.18·22-s + 0.432·23-s + 5.44·24-s + 0.454·25-s − 1.14·26-s − 0.120·27-s + 2.10·28-s − 1.49·29-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 \( 1 + 44.7T + 512T^{2} \)
3 \( 1 + 202.T + 1.96e4T^{2} \)
5 \( 1 + 1.68e3T + 1.95e6T^{2} \)
7 \( 1 - 4.58e3T + 4.03e7T^{2} \)
11 \( 1 + 2.91e4T + 2.35e9T^{2} \)
13 \( 1 - 5.96e4T + 1.06e10T^{2} \)
19 \( 1 - 3.93e5T + 3.22e11T^{2} \)
23 \( 1 - 5.81e5T + 1.80e12T^{2} \)
29 \( 1 + 5.69e6T + 1.45e13T^{2} \)
31 \( 1 + 5.57e5T + 2.64e13T^{2} \)
37 \( 1 - 2.31e6T + 1.29e14T^{2} \)
41 \( 1 + 2.59e7T + 3.27e14T^{2} \)
43 \( 1 - 1.10e7T + 5.02e14T^{2} \)
47 \( 1 + 1.90e7T + 1.11e15T^{2} \)
53 \( 1 + 8.95e7T + 3.29e15T^{2} \)
59 \( 1 - 1.61e8T + 8.66e15T^{2} \)
61 \( 1 + 8.62e7T + 1.16e16T^{2} \)
67 \( 1 + 7.28e7T + 2.72e16T^{2} \)
71 \( 1 - 2.46e8T + 4.58e16T^{2} \)
73 \( 1 - 3.69e7T + 5.88e16T^{2} \)
79 \( 1 + 5.32e8T + 1.19e17T^{2} \)
83 \( 1 - 3.39e8T + 1.86e17T^{2} \)
89 \( 1 - 4.69e8T + 3.50e17T^{2} \)
97 \( 1 - 6.49e6T + 7.60e17T^{2} \)
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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

−9.901951859720111783317663111784, −8.693035567385892163195690474179, −7.83644031789973008202239658515, −7.23893157560643698821354201261, −6.17281845666560874366203300360, −5.11810527880009624408940525118, −3.37220534923815080031211375667, −1.73035399283456835104211360334, −0.73282794485467704744124710269, 0, 0.73282794485467704744124710269, 1.73035399283456835104211360334, 3.37220534923815080031211375667, 5.11810527880009624408940525118, 6.17281845666560874366203300360, 7.23893157560643698821354201261, 7.83644031789973008202239658515, 8.693035567385892163195690474179, 9.901951859720111783317663111784

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