Properties

Label 2-3e2-1.1-c23-0-8
Degree $2$
Conductor $9$
Sign $-1$
Analytic cond. $30.1683$
Root an. cond. $5.49257$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.01e3·2-s + 7.74e6·4-s − 1.05e8·5-s + 3.81e9·7-s − 2.59e9·8-s − 4.21e11·10-s − 2.52e11·11-s − 3.59e12·13-s + 1.53e13·14-s − 7.53e13·16-s − 2.34e14·17-s − 6.23e14·19-s − 8.13e14·20-s − 1.01e15·22-s + 3.58e15·23-s − 8.82e14·25-s − 1.44e16·26-s + 2.95e16·28-s + 2.05e16·29-s + 1.36e17·31-s − 2.80e17·32-s − 9.40e17·34-s − 4.00e17·35-s − 1.23e18·37-s − 2.50e18·38-s + 2.72e17·40-s − 1.40e18·41-s + ⋯
L(s)  = 1  + 1.38·2-s + 0.922·4-s − 0.962·5-s + 0.728·7-s − 0.106·8-s − 1.33·10-s − 0.266·11-s − 0.555·13-s + 1.01·14-s − 1.07·16-s − 1.65·17-s − 1.22·19-s − 0.888·20-s − 0.369·22-s + 0.785·23-s − 0.0740·25-s − 0.770·26-s + 0.672·28-s + 0.313·29-s + 0.963·31-s − 1.37·32-s − 2.29·34-s − 0.701·35-s − 1.14·37-s − 1.70·38-s + 0.102·40-s − 0.397·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-1$
Analytic conductor: \(30.1683\)
Root analytic conductor: \(5.49257\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9,\ (\ :23/2),\ -1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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 \( 1 - 4.01e3T + 8.38e6T^{2} \)
5 \( 1 + 1.05e8T + 1.19e16T^{2} \)
7 \( 1 - 3.81e9T + 2.73e19T^{2} \)
11 \( 1 + 2.52e11T + 8.95e23T^{2} \)
13 \( 1 + 3.59e12T + 4.17e25T^{2} \)
17 \( 1 + 2.34e14T + 1.99e28T^{2} \)
19 \( 1 + 6.23e14T + 2.57e29T^{2} \)
23 \( 1 - 3.58e15T + 2.08e31T^{2} \)
29 \( 1 - 2.05e16T + 4.31e33T^{2} \)
31 \( 1 - 1.36e17T + 2.00e34T^{2} \)
37 \( 1 + 1.23e18T + 1.17e36T^{2} \)
41 \( 1 + 1.40e18T + 1.24e37T^{2} \)
43 \( 1 - 2.18e17T + 3.71e37T^{2} \)
47 \( 1 - 8.67e18T + 2.87e38T^{2} \)
53 \( 1 - 7.63e19T + 4.55e39T^{2} \)
59 \( 1 - 1.01e18T + 5.36e40T^{2} \)
61 \( 1 - 2.87e20T + 1.15e41T^{2} \)
67 \( 1 - 1.47e21T + 9.99e41T^{2} \)
71 \( 1 + 7.64e20T + 3.79e42T^{2} \)
73 \( 1 + 3.49e21T + 7.18e42T^{2} \)
79 \( 1 - 1.02e22T + 4.42e43T^{2} \)
83 \( 1 - 7.71e21T + 1.37e44T^{2} \)
89 \( 1 + 4.58e21T + 6.85e44T^{2} \)
97 \( 1 + 1.13e23T + 4.96e45T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.85840786671214873369200255648, −13.42206827315168010457399756975, −12.13508118028134002697742262804, −11.03131065402882068806585857207, −8.512632381608586760515883489243, −6.78775287522706808886968282456, −4.94991242423464308328279260265, −4.04637375498270240503524340812, −2.39776089745993745527527047428, 0, 2.39776089745993745527527047428, 4.04637375498270240503524340812, 4.94991242423464308328279260265, 6.78775287522706808886968282456, 8.512632381608586760515883489243, 11.03131065402882068806585857207, 12.13508118028134002697742262804, 13.42206827315168010457399756975, 14.85840786671214873369200255648

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