Properties

Label 2-72-1.1-c21-0-12
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  − 4.18e7·5-s − 6.87e8·7-s − 1.49e9·11-s − 8.84e9·13-s + 9.41e12·17-s − 4.23e13·19-s − 1.09e14·23-s + 1.27e15·25-s − 1.18e15·29-s − 1.95e15·31-s + 2.87e16·35-s + 4.20e16·37-s + 6.92e16·41-s + 8.95e16·43-s + 5.26e17·47-s − 8.63e16·49-s + 2.24e18·53-s + 6.26e16·55-s + 2.41e17·59-s + 3.17e18·61-s + 3.69e17·65-s − 2.04e19·67-s + 3.92e19·71-s − 4.75e19·73-s + 1.02e18·77-s + 8.63e18·79-s − 2.65e20·83-s + ⋯
L(s)  = 1  − 1.91·5-s − 0.919·7-s − 0.0174·11-s − 0.0178·13-s + 1.13·17-s − 1.58·19-s − 0.550·23-s + 2.66·25-s − 0.521·29-s − 0.427·31-s + 1.76·35-s + 1.43·37-s + 0.805·41-s + 0.631·43-s + 1.46·47-s − 0.154·49-s + 1.76·53-s + 0.0333·55-s + 0.0614·59-s + 0.570·61-s + 0.0340·65-s − 1.36·67-s + 1.42·71-s − 1.29·73-s + 0.0160·77-s + 0.102·79-s − 1.88·83-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 \( 1 + 4.18e7T + 4.76e14T^{2} \)
7 \( 1 + 6.87e8T + 5.58e17T^{2} \)
11 \( 1 + 1.49e9T + 7.40e21T^{2} \)
13 \( 1 + 8.84e9T + 2.47e23T^{2} \)
17 \( 1 - 9.41e12T + 6.90e25T^{2} \)
19 \( 1 + 4.23e13T + 7.14e26T^{2} \)
23 \( 1 + 1.09e14T + 3.94e28T^{2} \)
29 \( 1 + 1.18e15T + 5.13e30T^{2} \)
31 \( 1 + 1.95e15T + 2.08e31T^{2} \)
37 \( 1 - 4.20e16T + 8.55e32T^{2} \)
41 \( 1 - 6.92e16T + 7.38e33T^{2} \)
43 \( 1 - 8.95e16T + 2.00e34T^{2} \)
47 \( 1 - 5.26e17T + 1.30e35T^{2} \)
53 \( 1 - 2.24e18T + 1.62e36T^{2} \)
59 \( 1 - 2.41e17T + 1.54e37T^{2} \)
61 \( 1 - 3.17e18T + 3.10e37T^{2} \)
67 \( 1 + 2.04e19T + 2.22e38T^{2} \)
71 \( 1 - 3.92e19T + 7.52e38T^{2} \)
73 \( 1 + 4.75e19T + 1.34e39T^{2} \)
79 \( 1 - 8.63e18T + 7.08e39T^{2} \)
83 \( 1 + 2.65e20T + 1.99e40T^{2} \)
89 \( 1 - 4.57e20T + 8.65e40T^{2} \)
97 \( 1 + 9.21e20T + 5.27e41T^{2} \)
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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

−10.31023384459038135300362808246, −8.947694364758285209156872168572, −7.926420731474779103277423037093, −7.14339400132338150677540443602, −5.90542376969691656779667923418, −4.29714318652779134965632407566, −3.71502945632396885418102116334, −2.62211288398978119614309746948, −0.814314921929023694338795944310, 0, 0.814314921929023694338795944310, 2.62211288398978119614309746948, 3.71502945632396885418102116334, 4.29714318652779134965632407566, 5.90542376969691656779667923418, 7.14339400132338150677540443602, 7.926420731474779103277423037093, 8.947694364758285209156872168572, 10.31023384459038135300362808246

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