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Properties

Label 6-2e18-1.1-c21e3-0-0
Degree $6$
Conductor $262144$
Sign $1$
Analytic cond. $5.72242\times 10^{6}$
Root an. cond. $13.3740$
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  + 9.67e4·3-s + 2.41e7·5-s − 2.95e8·7-s − 1.58e9·9-s − 4.03e10·11-s − 1.33e11·13-s + 2.33e12·15-s + 7.79e12·17-s + 3.57e13·19-s − 2.86e13·21-s − 1.93e14·23-s + 1.39e14·25-s + 9.67e14·27-s − 5.60e15·29-s − 1.12e16·31-s − 3.90e15·33-s − 7.13e15·35-s + 2.42e16·37-s − 1.29e16·39-s − 2.98e17·41-s − 3.33e16·43-s − 3.82e16·45-s + 1.20e17·47-s − 1.21e18·49-s + 7.54e17·51-s + 1.13e18·53-s − 9.72e17·55-s + ⋯
L(s)  = 1  + 0.946·3-s + 1.10·5-s − 0.396·7-s − 0.151·9-s − 0.468·11-s − 0.269·13-s + 1.04·15-s + 0.938·17-s + 1.33·19-s − 0.374·21-s − 0.975·23-s + 0.291·25-s + 0.904·27-s − 2.47·29-s − 2.46·31-s − 0.443·33-s − 0.437·35-s + 0.829·37-s − 0.254·39-s − 3.46·41-s − 0.235·43-s − 0.167·45-s + 0.335·47-s − 2.18·49-s + 0.887·51-s + 0.894·53-s − 0.517·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $1$
Analytic conductor: \(5.72242\times 10^{6}\)
Root analytic conductor: \(13.3740\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 262144,\ (\ :21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(6.328936707\)
\(L(\frac12)\) \(\approx\) \(6.328936707\)
\(L(\frac{23}{2})\) 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 \( 1 \)
good3$S_4\times C_2$ \( 1 - 96764 T + 405548779 p^{3} T^{2} - 997170147832 p^{7} T^{3} + 405548779 p^{24} T^{4} - 96764 p^{42} T^{5} + p^{63} T^{6} \)
5$S_4\times C_2$ \( 1 - 24111774 T + 17686493631939 p^{2} T^{2} - 1220218992717725588 p^{4} T^{3} + 17686493631939 p^{23} T^{4} - 24111774 p^{42} T^{5} + p^{63} T^{6} \)
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

−9.589843085457167446852231226221, −8.990199353601002112510680226372, −8.902251186602030229591875223529, −8.529131059085488740012701402000, −7.80678743250795883093067663158, −7.74360052509911893587382734346, −7.48997504012487382125067647679, −6.81431385355753537057894564296, −6.54823189556116245535027698019, −6.18868573877925233816945266194, −5.39068130885793567884327437905, −5.38212333812453961656578564677, −5.33365663763305894078015021883, −4.75535408174990072870629119272, −3.85813281207135977803455434015, −3.62211529653262414451134201227, −3.43115602084896415945349773702, −3.08484287373745402821383928592, −2.59900439406679699509676028624, −2.03550980936006776015138166825, −1.92810032440921000534431125085, −1.72675285929642156341523166640, −1.07971355146546805195637249869, −0.51352504689340454909078765214, −0.34740440861731608347019356817, 0.34740440861731608347019356817, 0.51352504689340454909078765214, 1.07971355146546805195637249869, 1.72675285929642156341523166640, 1.92810032440921000534431125085, 2.03550980936006776015138166825, 2.59900439406679699509676028624, 3.08484287373745402821383928592, 3.43115602084896415945349773702, 3.62211529653262414451134201227, 3.85813281207135977803455434015, 4.75535408174990072870629119272, 5.33365663763305894078015021883, 5.38212333812453961656578564677, 5.39068130885793567884327437905, 6.18868573877925233816945266194, 6.54823189556116245535027698019, 6.81431385355753537057894564296, 7.48997504012487382125067647679, 7.74360052509911893587382734346, 7.80678743250795883093067663158, 8.529131059085488740012701402000, 8.902251186602030229591875223529, 8.990199353601002112510680226372, 9.589843085457167446852231226221

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