Properties

Label 2-72-1.1-c21-0-21
Degree $2$
Conductor $72$
Sign $-1$
Analytic cond. $201.223$
Root an. cond. $14.1853$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.08e7·5-s − 1.10e9·7-s − 1.54e11·11-s + 5.78e11·13-s − 1.59e12·17-s + 7.29e12·19-s + 1.56e14·23-s + 4.73e14·25-s + 1.80e15·29-s + 5.98e15·31-s − 3.40e16·35-s + 1.92e16·37-s + 1.01e17·41-s − 2.34e17·43-s − 1.31e17·47-s + 6.64e17·49-s − 2.51e18·53-s − 4.76e18·55-s − 4.37e18·59-s − 6.88e18·61-s + 1.78e19·65-s + 1.11e19·67-s + 4.02e19·71-s + 3.07e19·73-s + 1.71e20·77-s − 1.48e20·79-s − 9.43e19·83-s + ⋯
L(s)  = 1  + 1.41·5-s − 1.47·7-s − 1.79·11-s + 1.16·13-s − 0.192·17-s + 0.272·19-s + 0.786·23-s + 0.993·25-s + 0.798·29-s + 1.31·31-s − 2.08·35-s + 0.658·37-s + 1.18·41-s − 1.65·43-s − 0.364·47-s + 1.18·49-s − 1.97·53-s − 2.53·55-s − 1.11·59-s − 1.23·61-s + 1.64·65-s + 0.747·67-s + 1.46·71-s + 0.837·73-s + 2.66·77-s − 1.76·79-s − 0.667·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(201.223\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72,\ (\ :21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 3.08e7T + 4.76e14T^{2} \)
7 \( 1 + 1.10e9T + 5.58e17T^{2} \)
11 \( 1 + 1.54e11T + 7.40e21T^{2} \)
13 \( 1 - 5.78e11T + 2.47e23T^{2} \)
17 \( 1 + 1.59e12T + 6.90e25T^{2} \)
19 \( 1 - 7.29e12T + 7.14e26T^{2} \)
23 \( 1 - 1.56e14T + 3.94e28T^{2} \)
29 \( 1 - 1.80e15T + 5.13e30T^{2} \)
31 \( 1 - 5.98e15T + 2.08e31T^{2} \)
37 \( 1 - 1.92e16T + 8.55e32T^{2} \)
41 \( 1 - 1.01e17T + 7.38e33T^{2} \)
43 \( 1 + 2.34e17T + 2.00e34T^{2} \)
47 \( 1 + 1.31e17T + 1.30e35T^{2} \)
53 \( 1 + 2.51e18T + 1.62e36T^{2} \)
59 \( 1 + 4.37e18T + 1.54e37T^{2} \)
61 \( 1 + 6.88e18T + 3.10e37T^{2} \)
67 \( 1 - 1.11e19T + 2.22e38T^{2} \)
71 \( 1 - 4.02e19T + 7.52e38T^{2} \)
73 \( 1 - 3.07e19T + 1.34e39T^{2} \)
79 \( 1 + 1.48e20T + 7.08e39T^{2} \)
83 \( 1 + 9.43e19T + 1.99e40T^{2} \)
89 \( 1 - 2.18e20T + 8.65e40T^{2} \)
97 \( 1 - 1.66e20T + 5.27e41T^{2} \)
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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

−10.06582203636480352147847733537, −9.376829249834944413327819047051, −8.117045009820951616817698982711, −6.57218874841625418222689056474, −5.98387924237958456047333899551, −4.91369474270473215240611010031, −3.16071840479982716837370460929, −2.56360711184931768864016512725, −1.20473186547615320155234735811, 0, 1.20473186547615320155234735811, 2.56360711184931768864016512725, 3.16071840479982716837370460929, 4.91369474270473215240611010031, 5.98387924237958456047333899551, 6.57218874841625418222689056474, 8.117045009820951616817698982711, 9.376829249834944413327819047051, 10.06582203636480352147847733537

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