Properties

Label 2-17e2-1.1-c3-0-32
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  − 4.86·2-s + 5.06·3-s + 15.6·4-s − 18.5·5-s − 24.6·6-s + 14.0·7-s − 37.1·8-s − 1.34·9-s + 90.2·10-s + 19.4·11-s + 79.2·12-s + 29.9·13-s − 68.1·14-s − 94.0·15-s + 55.3·16-s + 6.51·18-s − 45.7·19-s − 290.·20-s + 71.0·21-s − 94.7·22-s − 89.3·23-s − 188.·24-s + 219.·25-s − 145.·26-s − 143.·27-s + 219.·28-s + 57.9·29-s + ⋯
L(s)  = 1  − 1.71·2-s + 0.974·3-s + 1.95·4-s − 1.66·5-s − 1.67·6-s + 0.757·7-s − 1.64·8-s − 0.0496·9-s + 2.85·10-s + 0.533·11-s + 1.90·12-s + 0.638·13-s − 1.30·14-s − 1.61·15-s + 0.865·16-s + 0.0853·18-s − 0.552·19-s − 3.24·20-s + 0.738·21-s − 0.917·22-s − 0.809·23-s − 1.59·24-s + 1.75·25-s − 1.09·26-s − 1.02·27-s + 1.47·28-s + 0.370·29-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 \( 1 + 4.86T + 8T^{2} \)
3 \( 1 - 5.06T + 27T^{2} \)
5 \( 1 + 18.5T + 125T^{2} \)
7 \( 1 - 14.0T + 343T^{2} \)
11 \( 1 - 19.4T + 1.33e3T^{2} \)
13 \( 1 - 29.9T + 2.19e3T^{2} \)
19 \( 1 + 45.7T + 6.85e3T^{2} \)
23 \( 1 + 89.3T + 1.21e4T^{2} \)
29 \( 1 - 57.9T + 2.43e4T^{2} \)
31 \( 1 - 161.T + 2.97e4T^{2} \)
37 \( 1 + 135.T + 5.06e4T^{2} \)
41 \( 1 + 56.8T + 6.89e4T^{2} \)
43 \( 1 - 52.1T + 7.95e4T^{2} \)
47 \( 1 + 482.T + 1.03e5T^{2} \)
53 \( 1 + 529.T + 1.48e5T^{2} \)
59 \( 1 + 280.T + 2.05e5T^{2} \)
61 \( 1 + 586.T + 2.26e5T^{2} \)
67 \( 1 - 367.T + 3.00e5T^{2} \)
71 \( 1 - 60.5T + 3.57e5T^{2} \)
73 \( 1 + 368.T + 3.89e5T^{2} \)
79 \( 1 - 217.T + 4.93e5T^{2} \)
83 \( 1 + 594.T + 5.71e5T^{2} \)
89 \( 1 + 887.T + 7.04e5T^{2} \)
97 \( 1 + 884.T + 9.12e5T^{2} \)
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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

−10.91796528548418094665129154006, −9.668636637659106458717764505712, −8.566200028034526911272207727419, −8.276248452313258501353254260724, −7.66639910099542424566098216272, −6.53664307639286199478214866067, −4.32927376484889706846712363410, −3.11079183814247211541913772885, −1.56615758347261809442965012993, 0, 1.56615758347261809442965012993, 3.11079183814247211541913772885, 4.32927376484889706846712363410, 6.53664307639286199478214866067, 7.66639910099542424566098216272, 8.276248452313258501353254260724, 8.566200028034526911272207727419, 9.668636637659106458717764505712, 10.91796528548418094665129154006

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