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Properties

Label 6-24e3-1.1-c21e3-0-2
Degree $6$
Conductor $13824$
Sign $-1$
Analytic cond. $301768.$
Root an. cond. $8.18990$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.77e5·3-s + 2.08e6·5-s − 1.20e9·7-s + 2.09e10·9-s + 1.38e10·11-s + 7.18e11·13-s − 3.68e11·15-s + 2.13e12·17-s − 4.01e13·19-s + 2.13e14·21-s + 2.78e14·23-s − 3.90e13·25-s − 2.05e15·27-s + 4.42e14·29-s + 8.01e15·31-s − 2.45e15·33-s − 2.50e15·35-s + 2.77e16·37-s − 1.27e17·39-s − 1.25e17·41-s − 2.29e17·43-s + 4.35e16·45-s − 4.48e17·47-s + 7.11e16·49-s − 3.78e17·51-s + 1.40e18·53-s + 2.87e16·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.0952·5-s − 1.61·7-s + 2·9-s + 0.160·11-s + 1.44·13-s − 0.164·15-s + 0.256·17-s − 1.50·19-s + 2.79·21-s + 1.40·23-s − 0.0819·25-s − 1.92·27-s + 0.195·29-s + 1.75·31-s − 0.278·33-s − 0.153·35-s + 0.948·37-s − 2.50·39-s − 1.46·41-s − 1.61·43-s + 0.190·45-s − 1.24·47-s + 0.127·49-s − 0.444·51-s + 1.10·53-s + 0.0153·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(13824\)    =    \(2^{9} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(301768.\)
Root analytic conductor: \(8.18990\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 13824,\ (\ :21/2, 21/2, 21/2),\ -1)\)

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$C_1$ \( ( 1 + p^{10} T )^{3} \)
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

−12.09344468568536851582970310246, −11.51447532228894117432890620804, −11.14965760894879747932707324007, −10.90441201662208846479299128515, −10.19236814942041903214563587048, −9.928240594890928701792752644719, −9.924014153411425710143387184409, −8.958808748928458335232701807835, −8.583146354264335176208896302346, −8.335302467683841981650666925753, −7.35235790111697782128860285239, −6.97647153336509637618572622222, −6.60182002609574735257649090827, −6.11706433279956963331425119331, −6.10509445143669605602281602265, −5.62189189686237243091434944827, −4.70466507627603774608404608474, −4.67388598607778752089276834617, −4.06636508542041307471171372788, −3.45902161990113917971209248954, −2.98258108472165282471093207779, −2.70027511312130115226258086909, −1.57567220201664377605531333671, −1.32041598908433861612565077638, −1.09689696765026389826204198252, 0, 0, 0, 1.09689696765026389826204198252, 1.32041598908433861612565077638, 1.57567220201664377605531333671, 2.70027511312130115226258086909, 2.98258108472165282471093207779, 3.45902161990113917971209248954, 4.06636508542041307471171372788, 4.67388598607778752089276834617, 4.70466507627603774608404608474, 5.62189189686237243091434944827, 6.10509445143669605602281602265, 6.11706433279956963331425119331, 6.60182002609574735257649090827, 6.97647153336509637618572622222, 7.35235790111697782128860285239, 8.335302467683841981650666925753, 8.583146354264335176208896302346, 8.958808748928458335232701807835, 9.924014153411425710143387184409, 9.928240594890928701792752644719, 10.19236814942041903214563587048, 10.90441201662208846479299128515, 11.14965760894879747932707324007, 11.51447532228894117432890620804, 12.09344468568536851582970310246

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