Properties

Label 2-17e2-1.1-c9-0-112
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  − 32.5·2-s + 213.·3-s + 547.·4-s − 2.40e3·5-s − 6.94e3·6-s − 8.34e3·7-s − 1.14e3·8-s + 2.58e4·9-s + 7.83e4·10-s + 2.94e4·11-s + 1.16e5·12-s + 1.14e5·13-s + 2.71e5·14-s − 5.13e5·15-s − 2.42e5·16-s − 8.40e5·18-s + 7.91e4·19-s − 1.31e6·20-s − 1.78e6·21-s − 9.57e5·22-s − 2.58e6·23-s − 2.43e5·24-s + 3.83e6·25-s − 3.73e6·26-s + 1.31e6·27-s − 4.56e6·28-s − 5.69e5·29-s + ⋯
L(s)  = 1  − 1.43·2-s + 1.52·3-s + 1.06·4-s − 1.72·5-s − 2.18·6-s − 1.31·7-s − 0.0986·8-s + 1.31·9-s + 2.47·10-s + 0.605·11-s + 1.62·12-s + 1.11·13-s + 1.89·14-s − 2.61·15-s − 0.926·16-s − 1.88·18-s + 0.139·19-s − 1.84·20-s − 1.99·21-s − 0.870·22-s − 1.92·23-s − 0.150·24-s + 1.96·25-s − 1.60·26-s + 0.475·27-s − 1.40·28-s − 0.149·29-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 \( 1 + 32.5T + 512T^{2} \)
3 \( 1 - 213.T + 1.96e4T^{2} \)
5 \( 1 + 2.40e3T + 1.95e6T^{2} \)
7 \( 1 + 8.34e3T + 4.03e7T^{2} \)
11 \( 1 - 2.94e4T + 2.35e9T^{2} \)
13 \( 1 - 1.14e5T + 1.06e10T^{2} \)
19 \( 1 - 7.91e4T + 3.22e11T^{2} \)
23 \( 1 + 2.58e6T + 1.80e12T^{2} \)
29 \( 1 + 5.69e5T + 1.45e13T^{2} \)
31 \( 1 - 3.28e6T + 2.64e13T^{2} \)
37 \( 1 + 1.35e7T + 1.29e14T^{2} \)
41 \( 1 - 5.82e6T + 3.27e14T^{2} \)
43 \( 1 - 4.20e7T + 5.02e14T^{2} \)
47 \( 1 + 2.37e7T + 1.11e15T^{2} \)
53 \( 1 - 4.78e7T + 3.29e15T^{2} \)
59 \( 1 - 1.36e8T + 8.66e15T^{2} \)
61 \( 1 - 8.58e7T + 1.16e16T^{2} \)
67 \( 1 - 1.38e8T + 2.72e16T^{2} \)
71 \( 1 + 1.34e7T + 4.58e16T^{2} \)
73 \( 1 - 1.85e8T + 5.88e16T^{2} \)
79 \( 1 + 9.57e7T + 1.19e17T^{2} \)
83 \( 1 - 6.11e8T + 1.86e17T^{2} \)
89 \( 1 - 3.05e8T + 3.50e17T^{2} \)
97 \( 1 + 4.08e8T + 7.60e17T^{2} \)
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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

−9.450672965795310462369870382455, −8.671990535589462371023236583780, −8.160200434071114309507210036887, −7.40348941697947766166158157842, −6.49215128796891097640579382669, −3.92546981265485495328231205965, −3.67579512206127220353986469152, −2.39264573341207868565406347818, −0.988869219977871400230099675997, 0, 0.988869219977871400230099675997, 2.39264573341207868565406347818, 3.67579512206127220353986469152, 3.92546981265485495328231205965, 6.49215128796891097640579382669, 7.40348941697947766166158157842, 8.160200434071114309507210036887, 8.671990535589462371023236583780, 9.450672965795310462369870382455

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