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Properties

Label 6-2e12-1.1-c25e3-0-1
Degree $6$
Conductor $4096$
Sign $-1$
Analytic cond. $254351.$
Root an. cond. $7.95986$
Motivic weight $25$
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  − 1.25e6·3-s + 4.85e7·5-s − 5.25e9·7-s − 2.89e11·9-s − 1.07e13·11-s + 1.14e14·13-s − 6.09e13·15-s − 1.01e15·17-s + 1.29e16·19-s + 6.59e15·21-s − 2.14e16·23-s − 1.06e17·25-s + 1.39e18·27-s + 2.28e18·29-s − 4.97e18·31-s + 1.34e19·33-s − 2.55e17·35-s + 1.06e20·37-s − 1.43e20·39-s + 2.14e20·41-s − 4.92e20·43-s − 1.40e19·45-s − 6.84e20·47-s − 1.83e21·49-s + 1.26e21·51-s − 1.10e22·53-s − 5.19e20·55-s + ⋯
L(s)  = 1  − 1.36·3-s + 0.0888·5-s − 0.143·7-s − 0.341·9-s − 1.02·11-s + 1.36·13-s − 0.121·15-s − 0.421·17-s + 1.34·19-s + 0.195·21-s − 0.204·23-s − 0.356·25-s + 1.78·27-s + 1.19·29-s − 1.13·31-s + 1.40·33-s − 0.0127·35-s + 2.65·37-s − 1.85·39-s + 1.48·41-s − 1.87·43-s − 0.0303·45-s − 0.859·47-s − 1.37·49-s + 0.574·51-s − 3.09·53-s − 0.0914·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+25/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $-1$
Analytic conductor: \(254351.\)
Root analytic conductor: \(7.95986\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 4096,\ (\ :25/2, 25/2, 25/2),\ -1)\)

Particular Values

\(L(13)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{27}{2})\) 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)$
bad2 \( 1 \)
good3$S_4\times C_2$ \( 1 + 1255436 T + 23026325833 p^{4} T^{2} + 1796261249099464 p^{6} T^{3} + 23026325833 p^{29} T^{4} + 1255436 p^{50} T^{5} + p^{75} T^{6} \)
5$S_4\times C_2$ \( 1 - 1940418 p^{2} T + 173898908306571 p^{4} T^{2} + \)\(29\!\cdots\!56\)\( p^{6} T^{3} + 173898908306571 p^{29} T^{4} - 1940418 p^{52} T^{5} + p^{75} T^{6} \)
7$S_4\times C_2$ \( 1 + 5257017240 T + \)\(26\!\cdots\!47\)\( p T^{2} + \)\(29\!\cdots\!76\)\( p^{4} T^{3} + \)\(26\!\cdots\!47\)\( p^{26} T^{4} + 5257017240 p^{50} T^{5} + p^{75} T^{6} \)
11$S_4\times C_2$ \( 1 + 10713764772516 T + \)\(21\!\cdots\!67\)\( p T^{2} + \)\(13\!\cdots\!96\)\( p^{3} T^{3} + \)\(21\!\cdots\!67\)\( p^{26} T^{4} + 10713764772516 p^{50} T^{5} + p^{75} T^{6} \)
13$S_4\times C_2$ \( 1 - 114437444085834 T + \)\(11\!\cdots\!43\)\( p T^{2} - \)\(52\!\cdots\!84\)\( p^{2} T^{3} + \)\(11\!\cdots\!43\)\( p^{26} T^{4} - 114437444085834 p^{50} T^{5} + p^{75} T^{6} \)
17$S_4\times C_2$ \( 1 + 1011476311341642 T + \)\(26\!\cdots\!43\)\( p T^{2} + \)\(50\!\cdots\!92\)\( p^{2} T^{3} + \)\(26\!\cdots\!43\)\( p^{26} T^{4} + 1011476311341642 p^{50} T^{5} + p^{75} T^{6} \)
19$S_4\times C_2$ \( 1 - 12956614226536644 T + \)\(97\!\cdots\!19\)\( p T^{2} - \)\(42\!\cdots\!48\)\( p^{2} T^{3} + \)\(97\!\cdots\!19\)\( p^{26} T^{4} - 12956614226536644 p^{50} T^{5} + p^{75} T^{6} \)
23$S_4\times C_2$ \( 1 + 933040317104760 p T + \)\(48\!\cdots\!13\)\( p^{2} T^{2} + \)\(33\!\cdots\!92\)\( p^{3} T^{3} + \)\(48\!\cdots\!13\)\( p^{27} T^{4} + 933040317104760 p^{51} T^{5} + p^{75} T^{6} \)
29$S_4\times C_2$ \( 1 - 2284613854595751162 T + \)\(12\!\cdots\!23\)\( T^{2} - \)\(16\!\cdots\!88\)\( T^{3} + \)\(12\!\cdots\!23\)\( p^{25} T^{4} - 2284613854595751162 p^{50} T^{5} + p^{75} T^{6} \)
31$S_4\times C_2$ \( 1 + 4973625711911164512 T + \)\(57\!\cdots\!93\)\( T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(57\!\cdots\!93\)\( p^{25} T^{4} + 4973625711911164512 p^{50} T^{5} + p^{75} T^{6} \)
37$S_4\times C_2$ \( 1 - \)\(10\!\cdots\!26\)\( T + \)\(71\!\cdots\!35\)\( T^{2} - \)\(31\!\cdots\!28\)\( T^{3} + \)\(71\!\cdots\!35\)\( p^{25} T^{4} - \)\(10\!\cdots\!26\)\( p^{50} T^{5} + p^{75} T^{6} \)
41$S_4\times C_2$ \( 1 - \)\(21\!\cdots\!38\)\( T + \)\(32\!\cdots\!83\)\( T^{2} - \)\(24\!\cdots\!92\)\( T^{3} + \)\(32\!\cdots\!83\)\( p^{25} T^{4} - \)\(21\!\cdots\!38\)\( p^{50} T^{5} + p^{75} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(49\!\cdots\!20\)\( T + \)\(27\!\cdots\!81\)\( T^{2} + \)\(70\!\cdots\!28\)\( T^{3} + \)\(27\!\cdots\!81\)\( p^{25} T^{4} + \)\(49\!\cdots\!20\)\( p^{50} T^{5} + p^{75} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(68\!\cdots\!48\)\( T + \)\(18\!\cdots\!89\)\( T^{2} + \)\(83\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!89\)\( p^{25} T^{4} + \)\(68\!\cdots\!48\)\( p^{50} T^{5} + p^{75} T^{6} \)
53$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!58\)\( T + \)\(70\!\cdots\!67\)\( T^{2} + \)\(30\!\cdots\!44\)\( T^{3} + \)\(70\!\cdots\!67\)\( p^{25} T^{4} + \)\(11\!\cdots\!58\)\( p^{50} T^{5} + p^{75} T^{6} \)
59$S_4\times C_2$ \( 1 + 59389977534775280988 p T + \)\(38\!\cdots\!77\)\( T^{2} + \)\(45\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!77\)\( p^{25} T^{4} + 59389977534775280988 p^{51} T^{5} + p^{75} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(51\!\cdots\!22\)\( T + \)\(19\!\cdots\!23\)\( T^{2} + \)\(44\!\cdots\!44\)\( T^{3} + \)\(19\!\cdots\!23\)\( p^{25} T^{4} + \)\(51\!\cdots\!22\)\( p^{50} T^{5} + p^{75} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!04\)\( T + \)\(59\!\cdots\!81\)\( T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(59\!\cdots\!81\)\( p^{25} T^{4} + \)\(12\!\cdots\!04\)\( p^{50} T^{5} + p^{75} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(46\!\cdots\!96\)\( T + \)\(94\!\cdots\!17\)\( T^{2} + \)\(13\!\cdots\!72\)\( T^{3} + \)\(94\!\cdots\!17\)\( p^{25} T^{4} + \)\(46\!\cdots\!96\)\( p^{50} T^{5} + p^{75} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(16\!\cdots\!90\)\( T + \)\(65\!\cdots\!47\)\( T^{2} - \)\(12\!\cdots\!24\)\( T^{3} + \)\(65\!\cdots\!47\)\( p^{25} T^{4} - \)\(16\!\cdots\!90\)\( p^{50} T^{5} + p^{75} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(14\!\cdots\!12\)\( T + \)\(13\!\cdots\!13\)\( T^{2} + \)\(84\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!13\)\( p^{25} T^{4} + \)\(14\!\cdots\!12\)\( p^{50} T^{5} + p^{75} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(56\!\cdots\!92\)\( T + \)\(28\!\cdots\!69\)\( T^{2} + \)\(10\!\cdots\!56\)\( T^{3} + \)\(28\!\cdots\!69\)\( p^{25} T^{4} + \)\(56\!\cdots\!92\)\( p^{50} T^{5} + p^{75} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(44\!\cdots\!74\)\( T + \)\(18\!\cdots\!91\)\( T^{2} + \)\(47\!\cdots\!08\)\( T^{3} + \)\(18\!\cdots\!91\)\( p^{25} T^{4} + \)\(44\!\cdots\!74\)\( p^{50} T^{5} + p^{75} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(65\!\cdots\!82\)\( T + \)\(13\!\cdots\!79\)\( T^{2} - \)\(58\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!79\)\( p^{25} T^{4} - \)\(65\!\cdots\!82\)\( p^{50} T^{5} + p^{75} T^{6} \)
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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

−12.33301828711530166341487288152, −11.53041129237724091947304773571, −11.39708877655975629323890636405, −11.33322336846387281774013909112, −10.74641604928053843953076948610, −10.21026873192485159702617047943, −9.856037825460185797642378949377, −9.220349612298761476875088449405, −8.820937351250073934385060568652, −8.263306593376812087131096757507, −7.70255756163437781383169386916, −7.55603832084513611088252781647, −6.64270738854759144656017080036, −6.07516070720785892741736556832, −6.02198793527623315676434403669, −5.72941108726958000938219935526, −4.98810894503482123500056266796, −4.62605580036687395262708512002, −4.29094618816929035218156880451, −3.24591494142526550025533292847, −2.92620727294403757572380935672, −2.89452938811350062322976936376, −1.79045528168563833029933880209, −1.28244378070271012269781881253, −1.12254850566463937529470812996, 0, 0, 0, 1.12254850566463937529470812996, 1.28244378070271012269781881253, 1.79045528168563833029933880209, 2.89452938811350062322976936376, 2.92620727294403757572380935672, 3.24591494142526550025533292847, 4.29094618816929035218156880451, 4.62605580036687395262708512002, 4.98810894503482123500056266796, 5.72941108726958000938219935526, 6.02198793527623315676434403669, 6.07516070720785892741736556832, 6.64270738854759144656017080036, 7.55603832084513611088252781647, 7.70255756163437781383169386916, 8.263306593376812087131096757507, 8.820937351250073934385060568652, 9.220349612298761476875088449405, 9.856037825460185797642378949377, 10.21026873192485159702617047943, 10.74641604928053843953076948610, 11.33322336846387281774013909112, 11.39708877655975629323890636405, 11.53041129237724091947304773571, 12.33301828711530166341487288152

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