Properties

Label 6-72e3-1.1-c21e3-0-2
Degree $6$
Conductor $373248$
Sign $-1$
Analytic cond. $8.14774\times 10^{6}$
Root an. cond. $14.1853$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.08e6·5-s − 1.20e9·7-s − 1.38e10·11-s + 7.18e11·13-s − 2.13e12·17-s − 4.01e13·19-s − 2.78e14·23-s − 3.90e13·25-s − 4.42e14·29-s + 8.01e15·31-s + 2.50e15·35-s + 2.77e16·37-s + 1.25e17·41-s − 2.29e17·43-s + 4.48e17·47-s + 7.11e16·49-s − 1.40e18·53-s + 2.87e16·55-s + 1.84e18·59-s − 3.29e18·61-s − 1.49e18·65-s − 3.34e19·67-s + 7.94e19·71-s − 4.66e19·73-s + 1.66e19·77-s − 1.97e20·79-s − 1.11e20·83-s + ⋯
L(s)  = 1  − 0.0952·5-s − 1.61·7-s − 0.160·11-s + 1.44·13-s − 0.256·17-s − 1.50·19-s − 1.40·23-s − 0.0819·25-s − 0.195·29-s + 1.75·31-s + 0.153·35-s + 0.948·37-s + 1.46·41-s − 1.61·43-s + 1.24·47-s + 0.127·49-s − 1.10·53-s + 0.0153·55-s + 0.469·59-s − 0.591·61-s − 0.137·65-s − 2.24·67-s + 2.89·71-s − 1.26·73-s + 0.259·77-s − 2.35·79-s − 0.791·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(373248\)    =    \(2^{9} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(8.14774\times 10^{6}\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 373248,\ (\ :21/2, 21/2, 21/2),\ -1)\)

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$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$S_4\times C_2$ \( 1 + 2080026 T + 1735894444323 p^{2} T^{2} + 21549971798269825532 p^{4} T^{3} + 1735894444323 p^{23} T^{4} + 2080026 p^{42} T^{5} + p^{63} T^{6} \)
7$S_4\times C_2$ \( 1 + 172183152 p T + 28195107577590405 p^{2} T^{2} + \)\(26\!\cdots\!72\)\( p^{3} T^{3} + 28195107577590405 p^{23} T^{4} + 172183152 p^{43} T^{5} + p^{63} T^{6} \)
11$S_4\times C_2$ \( 1 + 13839247500 T + 18571763310808269411 p T^{2} + \)\(14\!\cdots\!04\)\( p^{2} T^{3} + 18571763310808269411 p^{22} T^{4} + 13839247500 p^{42} T^{5} + p^{63} T^{6} \)
13$S_4\times C_2$ \( 1 - 718855551690 T + \)\(82\!\cdots\!07\)\( T^{2} - \)\(27\!\cdots\!48\)\( p T^{3} + \)\(82\!\cdots\!07\)\( p^{21} T^{4} - 718855551690 p^{42} T^{5} + p^{63} T^{6} \)
17$S_4\times C_2$ \( 1 + 2135189843046 T + \)\(67\!\cdots\!95\)\( p T^{2} + \)\(50\!\cdots\!72\)\( p^{2} T^{3} + \)\(67\!\cdots\!95\)\( p^{22} T^{4} + 2135189843046 p^{42} T^{5} + p^{63} T^{6} \)
19$S_4\times C_2$ \( 1 + 40122324686988 T + \)\(10\!\cdots\!03\)\( p T^{2} + \)\(15\!\cdots\!00\)\( p^{2} T^{3} + \)\(10\!\cdots\!03\)\( p^{22} T^{4} + 40122324686988 p^{42} T^{5} + p^{63} T^{6} \)
23$S_4\times C_2$ \( 1 + 278424417682632 T + \)\(86\!\cdots\!25\)\( T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + \)\(86\!\cdots\!25\)\( p^{21} T^{4} + 278424417682632 p^{42} T^{5} + p^{63} T^{6} \)
29$S_4\times C_2$ \( 1 + 442708167991794 T + \)\(12\!\cdots\!99\)\( T^{2} + \)\(77\!\cdots\!36\)\( p T^{3} + \)\(12\!\cdots\!99\)\( p^{21} T^{4} + 442708167991794 p^{42} T^{5} + p^{63} T^{6} \)
31$S_4\times C_2$ \( 1 - 8016070162990152 T + \)\(66\!\cdots\!13\)\( T^{2} - \)\(28\!\cdots\!24\)\( T^{3} + \)\(66\!\cdots\!13\)\( p^{21} T^{4} - 8016070162990152 p^{42} T^{5} + p^{63} T^{6} \)
37$S_4\times C_2$ \( 1 - 27729341388737058 T + \)\(13\!\cdots\!91\)\( T^{2} - \)\(20\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!91\)\( p^{21} T^{4} - 27729341388737058 p^{42} T^{5} + p^{63} T^{6} \)
41$S_4\times C_2$ \( 1 - 125648125186340562 T + \)\(21\!\cdots\!43\)\( T^{2} - \)\(15\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!43\)\( p^{21} T^{4} - 125648125186340562 p^{42} T^{5} + p^{63} T^{6} \)
43$S_4\times C_2$ \( 1 + 229052541499074612 T + \)\(51\!\cdots\!85\)\( T^{2} + \)\(74\!\cdots\!04\)\( T^{3} + \)\(51\!\cdots\!85\)\( p^{21} T^{4} + 229052541499074612 p^{42} T^{5} + p^{63} T^{6} \)
47$S_4\times C_2$ \( 1 - 448613782068047712 T + \)\(34\!\cdots\!81\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(34\!\cdots\!81\)\( p^{21} T^{4} - 448613782068047712 p^{42} T^{5} + p^{63} T^{6} \)
53$S_4\times C_2$ \( 1 + 1406206217208267066 T - \)\(48\!\cdots\!81\)\( T^{2} - \)\(18\!\cdots\!04\)\( T^{3} - \)\(48\!\cdots\!81\)\( p^{21} T^{4} + 1406206217208267066 p^{42} T^{5} + p^{63} T^{6} \)
59$S_4\times C_2$ \( 1 - 1844638981471622100 T + \)\(20\!\cdots\!69\)\( T^{2} - \)\(50\!\cdots\!68\)\( T^{3} + \)\(20\!\cdots\!69\)\( p^{21} T^{4} - 1844638981471622100 p^{42} T^{5} + p^{63} T^{6} \)
61$S_4\times C_2$ \( 1 + 3294066300350351382 T + \)\(69\!\cdots\!43\)\( T^{2} + \)\(21\!\cdots\!48\)\( T^{3} + \)\(69\!\cdots\!43\)\( p^{21} T^{4} + 3294066300350351382 p^{42} T^{5} + p^{63} T^{6} \)
67$S_4\times C_2$ \( 1 + 33491023693155020652 T + \)\(66\!\cdots\!61\)\( T^{2} + \)\(93\!\cdots\!32\)\( T^{3} + \)\(66\!\cdots\!61\)\( p^{21} T^{4} + 33491023693155020652 p^{42} T^{5} + p^{63} T^{6} \)
71$S_4\times C_2$ \( 1 - 79431018431598881160 T + \)\(38\!\cdots\!25\)\( T^{2} - \)\(11\!\cdots\!72\)\( T^{3} + \)\(38\!\cdots\!25\)\( p^{21} T^{4} - 79431018431598881160 p^{42} T^{5} + p^{63} T^{6} \)
73$S_4\times C_2$ \( 1 + 46612822906958319618 T + \)\(30\!\cdots\!27\)\( T^{2} + \)\(82\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!27\)\( p^{21} T^{4} + 46612822906958319618 p^{42} T^{5} + p^{63} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(19\!\cdots\!24\)\( T + \)\(28\!\cdots\!29\)\( T^{2} + \)\(27\!\cdots\!04\)\( T^{3} + \)\(28\!\cdots\!29\)\( p^{21} T^{4} + \)\(19\!\cdots\!24\)\( p^{42} T^{5} + p^{63} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!76\)\( T - \)\(44\!\cdots\!71\)\( T^{2} - \)\(21\!\cdots\!52\)\( p T^{3} - \)\(44\!\cdots\!71\)\( p^{21} T^{4} + \)\(11\!\cdots\!76\)\( p^{42} T^{5} + p^{63} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(73\!\cdots\!58\)\( T + \)\(39\!\cdots\!43\)\( T^{2} - \)\(13\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!43\)\( p^{21} T^{4} - \)\(73\!\cdots\!58\)\( p^{42} T^{5} + p^{63} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!22\)\( T + \)\(20\!\cdots\!87\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!87\)\( p^{21} T^{4} + \)\(15\!\cdots\!22\)\( p^{42} T^{5} + p^{63} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.06582203636480352147847733537, −9.376829249834944413327819047051, −9.062102360248290359317296426281, −8.947694364758285209156872168572, −8.173991681947699330774394010367, −8.117045009820951616817698982711, −7.926420731474779103277423037093, −7.14339400132338150677540443602, −6.77593525709775374052024815140, −6.57218874841625418222689056474, −5.98387924237958456047333899551, −5.94281136601750307735235540879, −5.90542376969691656779667923418, −4.91369474270473215240611010031, −4.56498557739984571892659630078, −4.29714318652779134965632407566, −3.71659645046990881627389106338, −3.71502945632396885418102116334, −3.16071840479982716837370460929, −2.62211288398978119614309746948, −2.56360711184931768864016512725, −2.05150015848198013902444283728, −1.44588857549832529339111158188, −1.20473186547615320155234735811, −0.814314921929023694338795944310, 0, 0, 0, 0.814314921929023694338795944310, 1.20473186547615320155234735811, 1.44588857549832529339111158188, 2.05150015848198013902444283728, 2.56360711184931768864016512725, 2.62211288398978119614309746948, 3.16071840479982716837370460929, 3.71502945632396885418102116334, 3.71659645046990881627389106338, 4.29714318652779134965632407566, 4.56498557739984571892659630078, 4.91369474270473215240611010031, 5.90542376969691656779667923418, 5.94281136601750307735235540879, 5.98387924237958456047333899551, 6.57218874841625418222689056474, 6.77593525709775374052024815140, 7.14339400132338150677540443602, 7.926420731474779103277423037093, 8.117045009820951616817698982711, 8.173991681947699330774394010367, 8.947694364758285209156872168572, 9.062102360248290359317296426281, 9.376829249834944413327819047051, 10.06582203636480352147847733537

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