Properties

Label 2-17e2-1.1-c9-0-136
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  − 7.00·2-s − 248.·3-s − 462.·4-s + 1.62e3·5-s + 1.74e3·6-s + 8.78e3·7-s + 6.83e3·8-s + 4.21e4·9-s − 1.13e4·10-s + 5.24e4·11-s + 1.15e5·12-s + 1.44e4·13-s − 6.15e4·14-s − 4.03e5·15-s + 1.89e5·16-s − 2.95e5·18-s − 9.31e5·19-s − 7.51e5·20-s − 2.18e6·21-s − 3.67e5·22-s − 1.37e6·23-s − 1.69e6·24-s + 6.85e5·25-s − 1.01e5·26-s − 5.57e6·27-s − 4.06e6·28-s + 4.42e6·29-s + ⋯
L(s)  = 1  − 0.309·2-s − 1.77·3-s − 0.904·4-s + 1.16·5-s + 0.548·6-s + 1.38·7-s + 0.589·8-s + 2.13·9-s − 0.360·10-s + 1.08·11-s + 1.60·12-s + 0.140·13-s − 0.428·14-s − 2.05·15-s + 0.721·16-s − 0.662·18-s − 1.64·19-s − 1.05·20-s − 2.44·21-s − 0.334·22-s − 1.02·23-s − 1.04·24-s + 0.351·25-s − 0.0434·26-s − 2.01·27-s − 1.24·28-s + 1.16·29-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 \( 1 + 7.00T + 512T^{2} \)
3 \( 1 + 248.T + 1.96e4T^{2} \)
5 \( 1 - 1.62e3T + 1.95e6T^{2} \)
7 \( 1 - 8.78e3T + 4.03e7T^{2} \)
11 \( 1 - 5.24e4T + 2.35e9T^{2} \)
13 \( 1 - 1.44e4T + 1.06e10T^{2} \)
19 \( 1 + 9.31e5T + 3.22e11T^{2} \)
23 \( 1 + 1.37e6T + 1.80e12T^{2} \)
29 \( 1 - 4.42e6T + 1.45e13T^{2} \)
31 \( 1 - 2.02e6T + 2.64e13T^{2} \)
37 \( 1 - 1.55e7T + 1.29e14T^{2} \)
41 \( 1 - 8.35e6T + 3.27e14T^{2} \)
43 \( 1 + 2.42e7T + 5.02e14T^{2} \)
47 \( 1 + 2.77e7T + 1.11e15T^{2} \)
53 \( 1 - 1.77e6T + 3.29e15T^{2} \)
59 \( 1 + 1.56e8T + 8.66e15T^{2} \)
61 \( 1 + 2.14e8T + 1.16e16T^{2} \)
67 \( 1 - 5.42e7T + 2.72e16T^{2} \)
71 \( 1 + 2.38e8T + 4.58e16T^{2} \)
73 \( 1 + 3.24e8T + 5.88e16T^{2} \)
79 \( 1 + 3.05e8T + 1.19e17T^{2} \)
83 \( 1 - 1.35e8T + 1.86e17T^{2} \)
89 \( 1 - 1.08e9T + 3.50e17T^{2} \)
97 \( 1 + 7.22e8T + 7.60e17T^{2} \)
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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

−10.01060322491729226551613693791, −8.960058932219276956716573607507, −7.898903783064640044336312686016, −6.38034368727241806343280544039, −5.91515456200709808891158152678, −4.70692100751197619843778513363, −4.38177946862741585287223279677, −1.74765619843517159730308181683, −1.19174248585652055877549941201, 0, 1.19174248585652055877549941201, 1.74765619843517159730308181683, 4.38177946862741585287223279677, 4.70692100751197619843778513363, 5.91515456200709808891158152678, 6.38034368727241806343280544039, 7.898903783064640044336312686016, 8.960058932219276956716573607507, 10.01060322491729226551613693791

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