Properties

Label 2-17e2-1.1-c3-0-39
Degree $2$
Conductor $289$
Sign $-1$
Analytic cond. $17.0515$
Root an. cond. $4.12935$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.70·2-s + 4.44·3-s − 5.09·4-s − 6.80·5-s − 7.57·6-s + 13.8·7-s + 22.3·8-s − 7.25·9-s + 11.6·10-s − 30.8·11-s − 22.6·12-s + 66.0·13-s − 23.6·14-s − 30.2·15-s + 2.65·16-s + 12.3·18-s − 79.7·19-s + 34.6·20-s + 61.6·21-s + 52.5·22-s − 28.8·23-s + 99.2·24-s − 78.6·25-s − 112.·26-s − 152.·27-s − 70.6·28-s − 266.·29-s + ⋯
L(s)  = 1  − 0.602·2-s + 0.855·3-s − 0.636·4-s − 0.609·5-s − 0.515·6-s + 0.749·7-s + 0.986·8-s − 0.268·9-s + 0.367·10-s − 0.845·11-s − 0.544·12-s + 1.40·13-s − 0.451·14-s − 0.520·15-s + 0.0414·16-s + 0.161·18-s − 0.962·19-s + 0.387·20-s + 0.640·21-s + 0.509·22-s − 0.261·23-s + 0.843·24-s − 0.629·25-s − 0.849·26-s − 1.08·27-s − 0.476·28-s − 1.70·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-1$
Analytic conductor: \(17.0515\)
Root analytic conductor: \(4.12935\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 289,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.70T + 8T^{2} \)
3 \( 1 - 4.44T + 27T^{2} \)
5 \( 1 + 6.80T + 125T^{2} \)
7 \( 1 - 13.8T + 343T^{2} \)
11 \( 1 + 30.8T + 1.33e3T^{2} \)
13 \( 1 - 66.0T + 2.19e3T^{2} \)
19 \( 1 + 79.7T + 6.85e3T^{2} \)
23 \( 1 + 28.8T + 1.21e4T^{2} \)
29 \( 1 + 266.T + 2.43e4T^{2} \)
31 \( 1 - 35.7T + 2.97e4T^{2} \)
37 \( 1 - 357.T + 5.06e4T^{2} \)
41 \( 1 + 32.7T + 6.89e4T^{2} \)
43 \( 1 + 516.T + 7.95e4T^{2} \)
47 \( 1 + 210.T + 1.03e5T^{2} \)
53 \( 1 - 87.2T + 1.48e5T^{2} \)
59 \( 1 + 310.T + 2.05e5T^{2} \)
61 \( 1 - 365.T + 2.26e5T^{2} \)
67 \( 1 + 660.T + 3.00e5T^{2} \)
71 \( 1 + 398.T + 3.57e5T^{2} \)
73 \( 1 + 643.T + 3.89e5T^{2} \)
79 \( 1 - 384.T + 4.93e5T^{2} \)
83 \( 1 + 153.T + 5.71e5T^{2} \)
89 \( 1 + 599.T + 7.04e5T^{2} \)
97 \( 1 + 44.5T + 9.12e5T^{2} \)
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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

−10.87439982539075284509575850819, −9.751337378713901950011558783271, −8.681910058503695006201109283211, −8.215956500431459307518663541968, −7.61540268124914233834787279672, −5.82283062917530429177169208901, −4.47425306285472296170645029836, −3.47986080197519134405289144752, −1.79327413601333073144987967281, 0, 1.79327413601333073144987967281, 3.47986080197519134405289144752, 4.47425306285472296170645029836, 5.82283062917530429177169208901, 7.61540268124914233834787279672, 8.215956500431459307518663541968, 8.681910058503695006201109283211, 9.751337378713901950011558783271, 10.87439982539075284509575850819

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