Properties

Label 2-17e2-1.1-c9-0-97
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  − 26.9·2-s − 20.4·3-s + 214.·4-s − 1.22e3·5-s + 549.·6-s + 4.89e3·7-s + 8.02e3·8-s − 1.92e4·9-s + 3.29e4·10-s − 9.03e4·11-s − 4.37e3·12-s + 1.14e5·13-s − 1.31e5·14-s + 2.49e4·15-s − 3.25e5·16-s + 5.19e5·18-s + 8.64e5·19-s − 2.61e5·20-s − 9.98e4·21-s + 2.43e6·22-s − 1.60e6·23-s − 1.63e5·24-s − 4.62e5·25-s − 3.07e6·26-s + 7.94e5·27-s + 1.04e6·28-s − 1.73e6·29-s + ⋯
L(s)  = 1  − 1.19·2-s − 0.145·3-s + 0.418·4-s − 0.873·5-s + 0.173·6-s + 0.770·7-s + 0.692·8-s − 0.978·9-s + 1.04·10-s − 1.86·11-s − 0.0608·12-s + 1.10·13-s − 0.917·14-s + 0.127·15-s − 1.24·16-s + 1.16·18-s + 1.52·19-s − 0.365·20-s − 0.112·21-s + 2.21·22-s − 1.19·23-s − 0.100·24-s − 0.236·25-s − 1.32·26-s + 0.287·27-s + 0.322·28-s − 0.456·29-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 \( 1 + 26.9T + 512T^{2} \)
3 \( 1 + 20.4T + 1.96e4T^{2} \)
5 \( 1 + 1.22e3T + 1.95e6T^{2} \)
7 \( 1 - 4.89e3T + 4.03e7T^{2} \)
11 \( 1 + 9.03e4T + 2.35e9T^{2} \)
13 \( 1 - 1.14e5T + 1.06e10T^{2} \)
19 \( 1 - 8.64e5T + 3.22e11T^{2} \)
23 \( 1 + 1.60e6T + 1.80e12T^{2} \)
29 \( 1 + 1.73e6T + 1.45e13T^{2} \)
31 \( 1 - 1.24e6T + 2.64e13T^{2} \)
37 \( 1 - 7.62e5T + 1.29e14T^{2} \)
41 \( 1 + 2.60e7T + 3.27e14T^{2} \)
43 \( 1 - 4.19e7T + 5.02e14T^{2} \)
47 \( 1 - 1.74e7T + 1.11e15T^{2} \)
53 \( 1 - 3.19e7T + 3.29e15T^{2} \)
59 \( 1 + 8.87e7T + 8.66e15T^{2} \)
61 \( 1 + 8.13e7T + 1.16e16T^{2} \)
67 \( 1 - 1.53e8T + 2.72e16T^{2} \)
71 \( 1 - 3.08e8T + 4.58e16T^{2} \)
73 \( 1 - 2.12e8T + 5.88e16T^{2} \)
79 \( 1 - 3.93e8T + 1.19e17T^{2} \)
83 \( 1 + 6.17e8T + 1.86e17T^{2} \)
89 \( 1 - 7.46e8T + 3.50e17T^{2} \)
97 \( 1 + 1.98e7T + 7.60e17T^{2} \)
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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

−9.715031279980229579017815758322, −8.530601108051015950592101315097, −7.991655201112768668905825920341, −7.50795765782985261859135808092, −5.78510211181149072024935987335, −4.86292668996052614824129775692, −3.52070157427525621621233327662, −2.17727297853061404327111467228, −0.838338546619125416258854437593, 0, 0.838338546619125416258854437593, 2.17727297853061404327111467228, 3.52070157427525621621233327662, 4.86292668996052614824129775692, 5.78510211181149072024935987335, 7.50795765782985261859135808092, 7.991655201112768668905825920341, 8.530601108051015950592101315097, 9.715031279980229579017815758322

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