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Properties

Label 6-72e3-1.1-c21e3-0-0
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41e7·5-s + 2.95e8·7-s + 4.03e10·11-s + 1.33e11·13-s − 7.79e12·17-s + 3.57e13·19-s − 1.93e14·23-s + 1.39e14·25-s − 5.60e15·29-s + 1.12e16·31-s + 7.13e15·35-s − 2.42e16·37-s + 2.98e17·41-s − 3.33e16·43-s + 1.20e17·47-s − 1.21e18·49-s + 1.13e18·53-s + 9.72e17·55-s − 9.22e18·59-s − 6.55e18·61-s + 3.22e18·65-s − 1.57e19·67-s + 4.11e19·71-s − 1.94e19·73-s + 1.19e19·77-s − 1.31e20·79-s − 6.40e19·83-s + ⋯
L(s)  = 1  + 1.10·5-s + 0.396·7-s + 0.468·11-s + 0.269·13-s − 0.938·17-s + 1.33·19-s − 0.975·23-s + 0.291·25-s − 2.47·29-s + 2.46·31-s + 0.437·35-s − 0.829·37-s + 3.46·41-s − 0.235·43-s + 0.335·47-s − 2.18·49-s + 0.894·53-s + 0.517·55-s − 2.34·59-s − 1.17·61-s + 0.297·65-s − 1.05·67-s + 1.49·71-s − 0.528·73-s + 0.185·77-s − 1.56·79-s − 0.452·83-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: \(0\)
Selberg data: \((6,\ 373248,\ (\ :21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(0.2440958713\)
\(L(\frac12)\) \(\approx\) \(0.2440958713\)
\(L(\frac{23}{2})\) 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$S_4\times C_2$ \( 1 - 24111774 T + 17686493631939 p^{2} T^{2} - 1220218992717725588 p^{4} T^{3} + 17686493631939 p^{23} T^{4} - 24111774 p^{42} T^{5} + p^{63} T^{6} \)
7$S_4\times C_2$ \( 1 - 42284040 p T + 3809985402511443 p^{3} T^{2} - \)\(86\!\cdots\!48\)\( p^{3} T^{3} + 3809985402511443 p^{24} T^{4} - 42284040 p^{43} T^{5} + p^{63} T^{6} \)
11$S_4\times C_2$ \( 1 - 40335108684 T + \)\(11\!\cdots\!17\)\( T^{2} - \)\(48\!\cdots\!64\)\( p T^{3} + \)\(11\!\cdots\!17\)\( p^{21} T^{4} - 40335108684 p^{42} T^{5} + p^{63} T^{6} \)
13$S_4\times C_2$ \( 1 - 133734425946 T + \)\(48\!\cdots\!63\)\( p T^{2} - \)\(42\!\cdots\!76\)\( p^{2} T^{3} + \)\(48\!\cdots\!63\)\( p^{22} T^{4} - 133734425946 p^{42} T^{5} + p^{63} T^{6} \)
17$S_4\times C_2$ \( 1 + 7797732274422 T + \)\(37\!\cdots\!71\)\( p^{4} T^{2} - \)\(85\!\cdots\!68\)\( p^{2} T^{3} + \)\(37\!\cdots\!71\)\( p^{25} T^{4} + 7797732274422 p^{42} T^{5} + p^{63} T^{6} \)
19$S_4\times C_2$ \( 1 - 35788199781996 T + \)\(12\!\cdots\!39\)\( p T^{2} - \)\(14\!\cdots\!92\)\( p^{2} T^{3} + \)\(12\!\cdots\!39\)\( p^{22} T^{4} - 35788199781996 p^{42} T^{5} + p^{63} T^{6} \)
23$S_4\times C_2$ \( 1 + 193770761479080 T + \)\(32\!\cdots\!17\)\( T^{2} + \)\(21\!\cdots\!56\)\( T^{3} + \)\(32\!\cdots\!17\)\( p^{21} T^{4} + 193770761479080 p^{42} T^{5} + p^{63} T^{6} \)
29$S_4\times C_2$ \( 1 + 5607343422466122 T + \)\(24\!\cdots\!83\)\( T^{2} + \)\(62\!\cdots\!88\)\( T^{3} + \)\(24\!\cdots\!83\)\( p^{21} T^{4} + 5607343422466122 p^{42} T^{5} + p^{63} T^{6} \)
31$S_4\times C_2$ \( 1 - 11246757871503072 T + \)\(97\!\cdots\!53\)\( T^{2} - \)\(49\!\cdots\!64\)\( T^{3} + \)\(97\!\cdots\!53\)\( p^{21} T^{4} - 11246757871503072 p^{42} T^{5} + p^{63} T^{6} \)
37$S_4\times C_2$ \( 1 + 24272499791100606 T + \)\(17\!\cdots\!75\)\( T^{2} + \)\(26\!\cdots\!28\)\( T^{3} + \)\(17\!\cdots\!75\)\( p^{21} T^{4} + 24272499791100606 p^{42} T^{5} + p^{63} T^{6} \)
41$S_4\times C_2$ \( 1 - 298159108991869602 T + \)\(47\!\cdots\!03\)\( T^{2} - \)\(50\!\cdots\!88\)\( T^{3} + \)\(47\!\cdots\!03\)\( p^{21} T^{4} - 298159108991869602 p^{42} T^{5} + p^{63} T^{6} \)
43$S_4\times C_2$ \( 1 + 33333932139754860 T + \)\(30\!\cdots\!21\)\( T^{2} + \)\(22\!\cdots\!68\)\( T^{3} + \)\(30\!\cdots\!21\)\( p^{21} T^{4} + 33333932139754860 p^{42} T^{5} + p^{63} T^{6} \)
47$S_4\times C_2$ \( 1 - 120874283547603888 T - \)\(88\!\cdots\!11\)\( T^{2} + \)\(47\!\cdots\!92\)\( T^{3} - \)\(88\!\cdots\!11\)\( p^{21} T^{4} - 120874283547603888 p^{42} T^{5} + p^{63} T^{6} \)
53$S_4\times C_2$ \( 1 - 1138443393004854222 T + \)\(47\!\cdots\!87\)\( T^{2} - \)\(34\!\cdots\!56\)\( T^{3} + \)\(47\!\cdots\!87\)\( p^{21} T^{4} - 1138443393004854222 p^{42} T^{5} + p^{63} T^{6} \)
59$S_4\times C_2$ \( 1 + 9225624498709937412 T + \)\(44\!\cdots\!77\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(44\!\cdots\!77\)\( p^{21} T^{4} + 9225624498709937412 p^{42} T^{5} + p^{63} T^{6} \)
61$S_4\times C_2$ \( 1 + 6554902294063924182 T + \)\(72\!\cdots\!43\)\( T^{2} + \)\(43\!\cdots\!64\)\( p T^{3} + \)\(72\!\cdots\!43\)\( p^{21} T^{4} + 6554902294063924182 p^{42} T^{5} + p^{63} T^{6} \)
67$S_4\times C_2$ \( 1 + 15793054074531629124 T + \)\(60\!\cdots\!01\)\( T^{2} + \)\(66\!\cdots\!68\)\( T^{3} + \)\(60\!\cdots\!01\)\( p^{21} T^{4} + 15793054074531629124 p^{42} T^{5} + p^{63} T^{6} \)
71$S_4\times C_2$ \( 1 - 41139582493467997704 T + \)\(13\!\cdots\!17\)\( T^{2} - \)\(27\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!17\)\( p^{21} T^{4} - 41139582493467997704 p^{42} T^{5} + p^{63} T^{6} \)
73$S_4\times C_2$ \( 1 + 19422167949903851970 T + \)\(37\!\cdots\!27\)\( T^{2} + \)\(49\!\cdots\!64\)\( T^{3} + \)\(37\!\cdots\!27\)\( p^{21} T^{4} + 19422167949903851970 p^{42} T^{5} + p^{63} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!88\)\( T + \)\(18\!\cdots\!73\)\( T^{2} + \)\(13\!\cdots\!04\)\( T^{3} + \)\(18\!\cdots\!73\)\( p^{21} T^{4} + \)\(13\!\cdots\!88\)\( p^{42} T^{5} + p^{63} T^{6} \)
83$S_4\times C_2$ \( 1 + 64013993832679681068 T + \)\(30\!\cdots\!69\)\( T^{2} + \)\(30\!\cdots\!64\)\( T^{3} + \)\(30\!\cdots\!69\)\( p^{21} T^{4} + 64013993832679681068 p^{42} T^{5} + p^{63} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(42\!\cdots\!66\)\( T + \)\(21\!\cdots\!31\)\( T^{2} + \)\(50\!\cdots\!72\)\( T^{3} + \)\(21\!\cdots\!31\)\( p^{21} T^{4} + \)\(42\!\cdots\!66\)\( p^{42} T^{5} + p^{63} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(32\!\cdots\!42\)\( T + \)\(10\!\cdots\!79\)\( T^{2} + \)\(44\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!79\)\( p^{21} T^{4} + \)\(32\!\cdots\!42\)\( p^{42} T^{5} + p^{63} T^{6} \)
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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

−9.555875921991535762009789280736, −8.953832552545590040366023283993, −8.712585718763667026146533992921, −8.190820405314183444288306552996, −7.77375010557620459885317962115, −7.45594805869319674188445065429, −7.30296757660894795095755671342, −6.50635782985139357177628329216, −6.31384603160852270927695272036, −6.04635110705038371080546240891, −5.72534826772087252621340762939, −5.22317553352015517732649851995, −5.02704929673258641833563406993, −4.32087559981441605241883336242, −4.14018377004506080324451053794, −3.97261159556828709571603902990, −3.05833843530439779117578168841, −2.93544947670646895612947363744, −2.68353203388035437608203798335, −1.82998475824443860127466662482, −1.82802370500270526100977500640, −1.61275167347292751596364029617, −0.943573727678738474526503437332, −0.794448246286980641991351496254, −0.05217144082355000123165536703, 0.05217144082355000123165536703, 0.794448246286980641991351496254, 0.943573727678738474526503437332, 1.61275167347292751596364029617, 1.82802370500270526100977500640, 1.82998475824443860127466662482, 2.68353203388035437608203798335, 2.93544947670646895612947363744, 3.05833843530439779117578168841, 3.97261159556828709571603902990, 4.14018377004506080324451053794, 4.32087559981441605241883336242, 5.02704929673258641833563406993, 5.22317553352015517732649851995, 5.72534826772087252621340762939, 6.04635110705038371080546240891, 6.31384603160852270927695272036, 6.50635782985139357177628329216, 7.30296757660894795095755671342, 7.45594805869319674188445065429, 7.77375010557620459885317962115, 8.190820405314183444288306552996, 8.712585718763667026146533992921, 8.953832552545590040366023283993, 9.555875921991535762009789280736

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