Properties

Label 6-3e6-1.1-c25e3-0-0
Degree $6$
Conductor $729$
Sign $1$
Analytic cond. $45269.1$
Root an. cond. $5.96990$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.67e3·2-s + 1.80e6·4-s + 1.63e8·5-s − 9.62e9·7-s + 3.82e10·8-s + 6.00e11·10-s + 5.94e12·11-s + 2.48e14·13-s − 3.53e13·14-s + 1.07e12·16-s − 6.64e15·17-s − 4.28e15·19-s + 2.94e14·20-s + 2.18e16·22-s + 5.45e16·23-s − 2.91e17·25-s + 9.12e17·26-s − 1.73e16·28-s + 5.06e14·29-s − 1.20e17·31-s + 2.38e18·32-s − 2.44e19·34-s − 1.56e18·35-s + 7.11e19·37-s − 1.57e19·38-s + 6.24e18·40-s + 1.78e20·41-s + ⋯
L(s)  = 1  + 0.634·2-s + 0.0538·4-s + 0.298·5-s − 0.262·7-s + 0.196·8-s + 0.189·10-s + 0.571·11-s + 2.95·13-s − 0.166·14-s + 0.000953·16-s − 2.76·17-s − 0.443·19-s + 0.0160·20-s + 0.362·22-s + 0.519·23-s − 0.976·25-s + 1.87·26-s − 0.0141·28-s + 0.000265·29-s − 0.0274·31-s + 0.365·32-s − 1.75·34-s − 0.0785·35-s + 1.77·37-s − 0.281·38-s + 0.0588·40-s + 1.23·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+25/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(45269.1\)
Root analytic conductor: \(5.96990\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 729,\ (\ :25/2, 25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(4.786255227\)
\(L(\frac12)\) \(\approx\) \(4.786255227\)
\(L(\frac{27}{2})\) 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)$
bad3 \( 1 \)
good2$S_4\times C_2$ \( 1 - 1839 p T + 366303 p^{5} T^{2} - 9121989 p^{13} T^{3} + 366303 p^{30} T^{4} - 1839 p^{51} T^{5} + p^{75} T^{6} \)
5$S_4\times C_2$ \( 1 - 1305222 p^{3} T + 508334131259403 p^{4} T^{2} - \)\(80\!\cdots\!56\)\( p^{6} T^{3} + 508334131259403 p^{29} T^{4} - 1305222 p^{53} T^{5} + p^{75} T^{6} \)
7$S_4\times C_2$ \( 1 + 9622572744 T + \)\(18\!\cdots\!95\)\( p T^{2} - \)\(81\!\cdots\!64\)\( p^{4} T^{3} + \)\(18\!\cdots\!95\)\( p^{26} T^{4} + 9622572744 p^{50} T^{5} + p^{75} T^{6} \)
11$S_4\times C_2$ \( 1 - 5946998130780 T + \)\(69\!\cdots\!61\)\( p^{2} T^{2} + \)\(35\!\cdots\!64\)\( p^{4} T^{3} + \)\(69\!\cdots\!61\)\( p^{27} T^{4} - 5946998130780 p^{50} T^{5} + p^{75} T^{6} \)
13$S_4\times C_2$ \( 1 - 248137774407690 T + \)\(36\!\cdots\!11\)\( T^{2} - \)\(26\!\cdots\!84\)\( p T^{3} + \)\(36\!\cdots\!11\)\( p^{25} T^{4} - 248137774407690 p^{50} T^{5} + p^{75} T^{6} \)
17$S_4\times C_2$ \( 1 + 6640885201245174 T + \)\(17\!\cdots\!39\)\( p T^{2} + \)\(29\!\cdots\!32\)\( p^{2} T^{3} + \)\(17\!\cdots\!39\)\( p^{26} T^{4} + 6640885201245174 p^{50} T^{5} + p^{75} T^{6} \)
19$S_4\times C_2$ \( 1 + 4282718959080516 T + \)\(11\!\cdots\!03\)\( p T^{2} + \)\(22\!\cdots\!88\)\( p^{2} T^{3} + \)\(11\!\cdots\!03\)\( p^{26} T^{4} + 4282718959080516 p^{50} T^{5} + p^{75} T^{6} \)
23$S_4\times C_2$ \( 1 - 2372128519633224 p T + \)\(26\!\cdots\!55\)\( p^{3} T^{2} - \)\(99\!\cdots\!56\)\( p^{3} T^{3} + \)\(26\!\cdots\!55\)\( p^{28} T^{4} - 2372128519633224 p^{51} T^{5} + p^{75} T^{6} \)
29$S_4\times C_2$ \( 1 - 506350856671782 T + \)\(37\!\cdots\!67\)\( T^{2} - \)\(69\!\cdots\!36\)\( T^{3} + \)\(37\!\cdots\!67\)\( p^{25} T^{4} - 506350856671782 p^{50} T^{5} + p^{75} T^{6} \)
31$S_4\times C_2$ \( 1 + 120595458951353856 T + \)\(39\!\cdots\!37\)\( T^{2} + \)\(20\!\cdots\!12\)\( T^{3} + \)\(39\!\cdots\!37\)\( p^{25} T^{4} + 120595458951353856 p^{50} T^{5} + p^{75} T^{6} \)
37$S_4\times C_2$ \( 1 - 1922271061255995498 p T + \)\(58\!\cdots\!95\)\( T^{2} - \)\(22\!\cdots\!04\)\( T^{3} + \)\(58\!\cdots\!95\)\( p^{25} T^{4} - 1922271061255995498 p^{51} T^{5} + p^{75} T^{6} \)
41$S_4\times C_2$ \( 1 - \)\(17\!\cdots\!26\)\( T + \)\(32\!\cdots\!87\)\( T^{2} - \)\(53\!\cdots\!72\)\( T^{3} + \)\(32\!\cdots\!87\)\( p^{25} T^{4} - \)\(17\!\cdots\!26\)\( p^{50} T^{5} + p^{75} T^{6} \)
43$S_4\times C_2$ \( 1 - \)\(23\!\cdots\!44\)\( T + \)\(20\!\cdots\!53\)\( T^{2} - \)\(30\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!53\)\( p^{25} T^{4} - \)\(23\!\cdots\!44\)\( p^{50} T^{5} + p^{75} T^{6} \)
47$S_4\times C_2$ \( 1 - \)\(57\!\cdots\!92\)\( T - \)\(89\!\cdots\!63\)\( T^{2} + \)\(50\!\cdots\!40\)\( T^{3} - \)\(89\!\cdots\!63\)\( p^{25} T^{4} - \)\(57\!\cdots\!92\)\( p^{50} T^{5} + p^{75} T^{6} \)
53$S_4\times C_2$ \( 1 + \)\(90\!\cdots\!58\)\( T + \)\(61\!\cdots\!15\)\( T^{2} + \)\(24\!\cdots\!68\)\( T^{3} + \)\(61\!\cdots\!15\)\( p^{25} T^{4} + \)\(90\!\cdots\!58\)\( p^{50} T^{5} + p^{75} T^{6} \)
59$S_4\times C_2$ \( 1 + \)\(23\!\cdots\!96\)\( T + \)\(71\!\cdots\!97\)\( T^{2} + \)\(91\!\cdots\!08\)\( T^{3} + \)\(71\!\cdots\!97\)\( p^{25} T^{4} + \)\(23\!\cdots\!96\)\( p^{50} T^{5} + p^{75} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!18\)\( T + \)\(79\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + \)\(79\!\cdots\!39\)\( p^{25} T^{4} + \)\(11\!\cdots\!18\)\( p^{50} T^{5} + p^{75} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(52\!\cdots\!24\)\( T + \)\(83\!\cdots\!13\)\( T^{2} - \)\(27\!\cdots\!48\)\( T^{3} + \)\(83\!\cdots\!13\)\( p^{25} T^{4} - \)\(52\!\cdots\!24\)\( p^{50} T^{5} + p^{75} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(26\!\cdots\!36\)\( T + \)\(74\!\cdots\!85\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!85\)\( p^{25} T^{4} + \)\(26\!\cdots\!36\)\( p^{50} T^{5} + p^{75} T^{6} \)
73$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!02\)\( T + \)\(53\!\cdots\!55\)\( T^{2} - \)\(27\!\cdots\!28\)\( T^{3} + \)\(53\!\cdots\!55\)\( p^{25} T^{4} + \)\(11\!\cdots\!02\)\( p^{50} T^{5} + p^{75} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(50\!\cdots\!60\)\( T + \)\(60\!\cdots\!97\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(60\!\cdots\!97\)\( p^{25} T^{4} + \)\(50\!\cdots\!60\)\( p^{50} T^{5} + p^{75} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(59\!\cdots\!84\)\( T + \)\(13\!\cdots\!09\)\( T^{2} - \)\(56\!\cdots\!96\)\( T^{3} + \)\(13\!\cdots\!09\)\( p^{25} T^{4} - \)\(59\!\cdots\!84\)\( p^{50} T^{5} + p^{75} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(33\!\cdots\!66\)\( T + \)\(13\!\cdots\!07\)\( T^{2} - \)\(31\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!07\)\( p^{25} T^{4} - \)\(33\!\cdots\!66\)\( p^{50} T^{5} + p^{75} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!66\)\( T + \)\(16\!\cdots\!23\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!23\)\( p^{25} T^{4} + \)\(12\!\cdots\!66\)\( p^{50} T^{5} + p^{75} T^{6} \)
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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

−13.43111455840707688865728528112, −12.86683805926877719578025449753, −12.66587085628942664878716721025, −11.59038117103746022687355880482, −11.25166433595518357067832733635, −10.85495472046424028287799765993, −10.71352486519984034920197962172, −9.554832213248793175167619404283, −9.139544031113178198750872578739, −8.922667313366115424906628480138, −8.189672218821028325517795210232, −7.76985995529455777981801587896, −6.67044891104854575298041547616, −6.55683577547876976356563499475, −5.89552407288953790840258798953, −5.85296802424939885154673520510, −4.49268144646890002100955956352, −4.29545692114996162540600878747, −4.14613679868101772296265438649, −3.14817286772350203942095806573, −2.85893497144167196076417985202, −1.82119116541440768349269492344, −1.71318197304510312544595560670, −0.945506555640721259234792976530, −0.37994093876519177585829330473, 0.37994093876519177585829330473, 0.945506555640721259234792976530, 1.71318197304510312544595560670, 1.82119116541440768349269492344, 2.85893497144167196076417985202, 3.14817286772350203942095806573, 4.14613679868101772296265438649, 4.29545692114996162540600878747, 4.49268144646890002100955956352, 5.85296802424939885154673520510, 5.89552407288953790840258798953, 6.55683577547876976356563499475, 6.67044891104854575298041547616, 7.76985995529455777981801587896, 8.189672218821028325517795210232, 8.922667313366115424906628480138, 9.139544031113178198750872578739, 9.554832213248793175167619404283, 10.71352486519984034920197962172, 10.85495472046424028287799765993, 11.25166433595518357067832733635, 11.59038117103746022687355880482, 12.66587085628942664878716721025, 12.86683805926877719578025449753, 13.43111455840707688865728528112

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