Properties

Label 2-17e2-1.1-c9-0-23
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.91·2-s + 33.9·3-s − 432.·4-s − 34.8·5-s − 302.·6-s − 8.53e3·7-s + 8.42e3·8-s − 1.85e4·9-s + 310.·10-s − 828.·11-s − 1.46e4·12-s + 1.47e5·13-s + 7.61e4·14-s − 1.18e3·15-s + 1.46e5·16-s + 1.65e5·18-s + 3.85e5·19-s + 1.50e4·20-s − 2.89e5·21-s + 7.39e3·22-s − 1.98e6·23-s + 2.85e5·24-s − 1.95e6·25-s − 1.31e6·26-s − 1.29e6·27-s + 3.69e6·28-s − 2.63e6·29-s + ⋯
L(s)  = 1  − 0.394·2-s + 0.241·3-s − 0.844·4-s − 0.0249·5-s − 0.0953·6-s − 1.34·7-s + 0.726·8-s − 0.941·9-s + 0.00983·10-s − 0.0170·11-s − 0.204·12-s + 1.43·13-s + 0.529·14-s − 0.00603·15-s + 0.558·16-s + 0.371·18-s + 0.679·19-s + 0.0210·20-s − 0.325·21-s + 0.00672·22-s − 1.47·23-s + 0.175·24-s − 0.999·25-s − 0.564·26-s − 0.469·27-s + 1.13·28-s − 0.692·29-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.3586137114\)
\(L(\frac12)\) \(\approx\) \(0.3586137114\)
\(L(\frac{11}{2})\) 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 \( 1 + 8.91T + 512T^{2} \)
3 \( 1 - 33.9T + 1.96e4T^{2} \)
5 \( 1 + 34.8T + 1.95e6T^{2} \)
7 \( 1 + 8.53e3T + 4.03e7T^{2} \)
11 \( 1 + 828.T + 2.35e9T^{2} \)
13 \( 1 - 1.47e5T + 1.06e10T^{2} \)
19 \( 1 - 3.85e5T + 3.22e11T^{2} \)
23 \( 1 + 1.98e6T + 1.80e12T^{2} \)
29 \( 1 + 2.63e6T + 1.45e13T^{2} \)
31 \( 1 + 6.41e6T + 2.64e13T^{2} \)
37 \( 1 + 1.89e7T + 1.29e14T^{2} \)
41 \( 1 - 9.32e6T + 3.27e14T^{2} \)
43 \( 1 + 1.72e7T + 5.02e14T^{2} \)
47 \( 1 - 1.46e7T + 1.11e15T^{2} \)
53 \( 1 + 9.96e7T + 3.29e15T^{2} \)
59 \( 1 - 7.35e7T + 8.66e15T^{2} \)
61 \( 1 - 6.43e5T + 1.16e16T^{2} \)
67 \( 1 + 2.31e8T + 2.72e16T^{2} \)
71 \( 1 + 4.38e7T + 4.58e16T^{2} \)
73 \( 1 - 2.09e8T + 5.88e16T^{2} \)
79 \( 1 - 7.18e7T + 1.19e17T^{2} \)
83 \( 1 + 3.97e8T + 1.86e17T^{2} \)
89 \( 1 - 2.63e8T + 3.50e17T^{2} \)
97 \( 1 + 3.00e8T + 7.60e17T^{2} \)
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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

−9.961529775059738651497492188118, −9.244474085877715287767370225376, −8.531824188989155823289779964754, −7.59975367927648180288631127610, −6.20029014974095326705463583739, −5.49452805925038345678733614130, −3.82691685336361379525898547477, −3.35140588625553971732155343460, −1.72290857506689629737689675038, −0.27686063198420554002498091343, 0.27686063198420554002498091343, 1.72290857506689629737689675038, 3.35140588625553971732155343460, 3.82691685336361379525898547477, 5.49452805925038345678733614130, 6.20029014974095326705463583739, 7.59975367927648180288631127610, 8.531824188989155823289779964754, 9.244474085877715287767370225376, 9.961529775059738651497492188118

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