Properties

Label 2-17e2-1.1-c3-0-38
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  − 0.820·2-s + 0.130·3-s − 7.32·4-s + 8.70·5-s − 0.106·6-s + 11.4·7-s + 12.5·8-s − 26.9·9-s − 7.14·10-s − 7.27·11-s − 0.952·12-s − 52.1·13-s − 9.35·14-s + 1.13·15-s + 48.3·16-s + 22.1·18-s + 84.1·19-s − 63.7·20-s + 1.48·21-s + 5.96·22-s + 157.·23-s + 1.63·24-s − 49.1·25-s + 42.7·26-s − 7.01·27-s − 83.5·28-s − 276.·29-s + ⋯
L(s)  = 1  − 0.290·2-s + 0.0250·3-s − 0.915·4-s + 0.778·5-s − 0.00725·6-s + 0.615·7-s + 0.555·8-s − 0.999·9-s − 0.225·10-s − 0.199·11-s − 0.0229·12-s − 1.11·13-s − 0.178·14-s + 0.0194·15-s + 0.754·16-s + 0.289·18-s + 1.01·19-s − 0.713·20-s + 0.0154·21-s + 0.0578·22-s + 1.42·23-s + 0.0139·24-s − 0.393·25-s + 0.322·26-s − 0.0500·27-s − 0.563·28-s − 1.76·29-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 \( 1 + 0.820T + 8T^{2} \)
3 \( 1 - 0.130T + 27T^{2} \)
5 \( 1 - 8.70T + 125T^{2} \)
7 \( 1 - 11.4T + 343T^{2} \)
11 \( 1 + 7.27T + 1.33e3T^{2} \)
13 \( 1 + 52.1T + 2.19e3T^{2} \)
19 \( 1 - 84.1T + 6.85e3T^{2} \)
23 \( 1 - 157.T + 1.21e4T^{2} \)
29 \( 1 + 276.T + 2.43e4T^{2} \)
31 \( 1 + 235.T + 2.97e4T^{2} \)
37 \( 1 + 352.T + 5.06e4T^{2} \)
41 \( 1 + 90.2T + 6.89e4T^{2} \)
43 \( 1 + 167.T + 7.95e4T^{2} \)
47 \( 1 + 589.T + 1.03e5T^{2} \)
53 \( 1 - 288.T + 1.48e5T^{2} \)
59 \( 1 - 519.T + 2.05e5T^{2} \)
61 \( 1 + 557.T + 2.26e5T^{2} \)
67 \( 1 - 304.T + 3.00e5T^{2} \)
71 \( 1 - 316.T + 3.57e5T^{2} \)
73 \( 1 - 66.6T + 3.89e5T^{2} \)
79 \( 1 + 118.T + 4.93e5T^{2} \)
83 \( 1 + 215.T + 5.71e5T^{2} \)
89 \( 1 - 931.T + 7.04e5T^{2} \)
97 \( 1 - 402.T + 9.12e5T^{2} \)
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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

−10.80970479432603838388414012837, −9.691485166832703605977766112965, −9.158111358158718752664643536259, −8.145122610647729335881771184148, −7.16704576104983263877089838831, −5.39702332383909838719184423683, −5.14357683494766949323042401044, −3.37916915246236202328194421329, −1.79076213772651482353341510891, 0, 1.79076213772651482353341510891, 3.37916915246236202328194421329, 5.14357683494766949323042401044, 5.39702332383909838719184423683, 7.16704576104983263877089838831, 8.145122610647729335881771184148, 9.158111358158718752664643536259, 9.691485166832703605977766112965, 10.80970479432603838388414012837

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