Properties

Label 2-3e2-1.1-c67-0-18
Degree $2$
Conductor $9$
Sign $-1$
Analytic cond. $255.861$
Root an. cond. $15.9956$
Motivic weight $67$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.26e10·2-s + 3.67e20·4-s + 1.61e23·5-s + 2.93e28·7-s − 4.99e30·8-s − 3.65e33·10-s − 1.03e35·11-s − 9.30e35·13-s − 6.66e38·14-s + 5.90e40·16-s − 2.52e41·17-s − 4.10e41·19-s + 5.92e43·20-s + 2.36e45·22-s + 7.28e44·23-s − 4.17e46·25-s + 2.11e46·26-s + 1.07e49·28-s + 8.66e48·29-s + 5.42e49·31-s − 6.03e50·32-s + 5.72e51·34-s + 4.73e51·35-s + 2.44e52·37-s + 9.32e51·38-s − 8.04e53·40-s − 8.84e52·41-s + ⋯
L(s)  = 1  − 1.86·2-s + 2.49·4-s + 0.619·5-s + 1.43·7-s − 2.78·8-s − 1.15·10-s − 1.34·11-s − 0.0448·13-s − 2.68·14-s + 2.71·16-s − 1.51·17-s − 0.0596·19-s + 1.54·20-s + 2.52·22-s + 0.175·23-s − 0.616·25-s + 0.0837·26-s + 3.57·28-s + 0.886·29-s + 0.594·31-s − 2.28·32-s + 2.83·34-s + 0.889·35-s + 0.712·37-s + 0.111·38-s − 1.72·40-s − 0.0829·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+67/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-1$
Analytic conductor: \(255.861\)
Root analytic conductor: \(15.9956\)
Motivic weight: \(67\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9,\ (\ :67/2),\ -1)\)

Particular Values

\(L(34)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{69}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 2.26e10T + 1.47e20T^{2} \)
5 \( 1 - 1.61e23T + 6.77e46T^{2} \)
7 \( 1 - 2.93e28T + 4.18e56T^{2} \)
11 \( 1 + 1.03e35T + 5.93e69T^{2} \)
13 \( 1 + 9.30e35T + 4.30e74T^{2} \)
17 \( 1 + 2.52e41T + 2.75e82T^{2} \)
19 \( 1 + 4.10e41T + 4.74e85T^{2} \)
23 \( 1 - 7.28e44T + 1.72e91T^{2} \)
29 \( 1 - 8.66e48T + 9.56e97T^{2} \)
31 \( 1 - 5.42e49T + 8.34e99T^{2} \)
37 \( 1 - 2.44e52T + 1.17e105T^{2} \)
41 \( 1 + 8.84e52T + 1.13e108T^{2} \)
43 \( 1 + 3.35e54T + 2.76e109T^{2} \)
47 \( 1 - 1.34e56T + 1.07e112T^{2} \)
53 \( 1 - 3.87e57T + 3.36e115T^{2} \)
59 \( 1 - 2.62e59T + 4.43e118T^{2} \)
61 \( 1 + 4.77e59T + 4.14e119T^{2} \)
67 \( 1 + 2.36e61T + 2.22e122T^{2} \)
71 \( 1 + 2.36e61T + 1.08e124T^{2} \)
73 \( 1 - 7.86e61T + 6.96e124T^{2} \)
79 \( 1 + 2.74e63T + 1.38e127T^{2} \)
83 \( 1 + 9.16e63T + 3.78e128T^{2} \)
89 \( 1 - 3.36e65T + 4.06e130T^{2} \)
97 \( 1 - 6.50e66T + 1.29e133T^{2} \)
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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

−10.14223790159457849409322887570, −8.874844805520706524159872466096, −8.149252440867280361816425317787, −7.26814248679887920449718494250, −6.00699427613460170744990652437, −4.73492720070418793023645048572, −2.54911642168043193285587331131, −2.04930748793261219294132245389, −1.04274823478568310987206575717, 0, 1.04274823478568310987206575717, 2.04930748793261219294132245389, 2.54911642168043193285587331131, 4.73492720070418793023645048572, 6.00699427613460170744990652437, 7.26814248679887920449718494250, 8.149252440867280361816425317787, 8.874844805520706524159872466096, 10.14223790159457849409322887570

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