Properties

Label 2-17e2-1.1-c9-0-62
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  + 2.64·2-s − 62.7·3-s − 504.·4-s − 1.95e3·5-s − 166.·6-s − 4.15e3·7-s − 2.69e3·8-s − 1.57e4·9-s − 5.18e3·10-s − 7.30e4·11-s + 3.16e4·12-s − 5.43e4·13-s − 1.09e4·14-s + 1.22e5·15-s + 2.51e5·16-s − 4.17e4·18-s + 8.11e5·19-s + 9.88e5·20-s + 2.60e5·21-s − 1.93e5·22-s − 1.63e5·23-s + 1.68e5·24-s + 1.87e6·25-s − 1.43e5·26-s + 2.22e6·27-s + 2.09e6·28-s + 5.42e6·29-s + ⋯
L(s)  = 1  + 0.117·2-s − 0.446·3-s − 0.986·4-s − 1.40·5-s − 0.0523·6-s − 0.653·7-s − 0.232·8-s − 0.800·9-s − 0.163·10-s − 1.50·11-s + 0.440·12-s − 0.527·13-s − 0.0764·14-s + 0.626·15-s + 0.959·16-s − 0.0936·18-s + 1.42·19-s + 1.38·20-s + 0.292·21-s − 0.176·22-s − 0.121·23-s + 0.103·24-s + 0.962·25-s − 0.0617·26-s + 0.804·27-s + 0.644·28-s + 1.42·29-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 \( 1 - 2.64T + 512T^{2} \)
3 \( 1 + 62.7T + 1.96e4T^{2} \)
5 \( 1 + 1.95e3T + 1.95e6T^{2} \)
7 \( 1 + 4.15e3T + 4.03e7T^{2} \)
11 \( 1 + 7.30e4T + 2.35e9T^{2} \)
13 \( 1 + 5.43e4T + 1.06e10T^{2} \)
19 \( 1 - 8.11e5T + 3.22e11T^{2} \)
23 \( 1 + 1.63e5T + 1.80e12T^{2} \)
29 \( 1 - 5.42e6T + 1.45e13T^{2} \)
31 \( 1 + 9.46e5T + 2.64e13T^{2} \)
37 \( 1 + 1.91e7T + 1.29e14T^{2} \)
41 \( 1 + 2.07e7T + 3.27e14T^{2} \)
43 \( 1 + 3.17e7T + 5.02e14T^{2} \)
47 \( 1 - 4.92e7T + 1.11e15T^{2} \)
53 \( 1 - 6.39e7T + 3.29e15T^{2} \)
59 \( 1 + 1.75e7T + 8.66e15T^{2} \)
61 \( 1 - 1.01e8T + 1.16e16T^{2} \)
67 \( 1 - 5.10e7T + 2.72e16T^{2} \)
71 \( 1 - 1.39e8T + 4.58e16T^{2} \)
73 \( 1 + 1.70e8T + 5.88e16T^{2} \)
79 \( 1 - 4.92e8T + 1.19e17T^{2} \)
83 \( 1 - 3.76e8T + 1.86e17T^{2} \)
89 \( 1 + 8.85e8T + 3.50e17T^{2} \)
97 \( 1 - 1.16e8T + 7.60e17T^{2} \)
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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

−9.865432995621979860977415867780, −8.595546180484645368977056741378, −7.992240324451732433824098008265, −6.95231985060456455805073667777, −5.42296273529856655908171132706, −4.93304278761790923034495731701, −3.60288490515569940008371493407, −2.89301131596381083644980457447, −0.61561882790655352670497234899, 0, 0.61561882790655352670497234899, 2.89301131596381083644980457447, 3.60288490515569940008371493407, 4.93304278761790923034495731701, 5.42296273529856655908171132706, 6.95231985060456455805073667777, 7.992240324451732433824098008265, 8.595546180484645368977056741378, 9.865432995621979860977415867780

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