Properties

Label 6-80e3-1.1-c9e3-0-0
Degree $6$
Conductor $512000$
Sign $1$
Analytic cond. $69949.1$
Root an. cond. $6.41894$
Motivic weight $9$
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  − 84·3-s + 1.87e3·5-s + 5.52e3·7-s − 2.45e3·9-s − 5.55e3·11-s − 8.30e4·13-s − 1.57e5·15-s + 3.67e5·17-s − 1.48e6·19-s − 4.63e5·21-s + 4.99e5·23-s + 2.34e6·25-s − 1.61e6·27-s + 5.23e6·29-s − 1.27e7·31-s + 4.66e5·33-s + 1.03e7·35-s + 2.17e7·37-s + 6.97e6·39-s + 2.74e7·41-s + 2.32e7·43-s − 4.60e6·45-s + 2.87e7·47-s − 3.17e7·49-s − 3.08e7·51-s − 4.56e7·53-s − 1.04e7·55-s + ⋯
L(s)  = 1  − 0.598·3-s + 1.34·5-s + 0.868·7-s − 0.124·9-s − 0.114·11-s − 0.806·13-s − 0.803·15-s + 1.06·17-s − 2.62·19-s − 0.520·21-s + 0.372·23-s + 6/5·25-s − 0.583·27-s + 1.37·29-s − 2.47·31-s + 0.0685·33-s + 1.16·35-s + 1.90·37-s + 0.483·39-s + 1.51·41-s + 1.03·43-s − 0.167·45-s + 0.857·47-s − 0.786·49-s − 0.638·51-s − 0.794·53-s − 0.153·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512000 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(512000\)    =    \(2^{12} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(69949.1\)
Root analytic conductor: \(6.41894\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 512000,\ (\ :9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(4.825822051\)
\(L(\frac12)\) \(\approx\) \(4.825822051\)
\(L(\frac{11}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - p^{4} T )^{3} \)
good3$S_4\times C_2$ \( 1 + 28 p T + 1057 p^{2} T^{2} + 96872 p^{3} T^{3} + 1057 p^{11} T^{4} + 28 p^{19} T^{5} + p^{27} T^{6} \)
7$S_4\times C_2$ \( 1 - 5520 T + 8888307 p T^{2} - 6697236064 p^{2} T^{3} + 8888307 p^{10} T^{4} - 5520 p^{18} T^{5} + p^{27} T^{6} \)
11$S_4\times C_2$ \( 1 + 5556 T + 2594856177 T^{2} + 72877014191416 T^{3} + 2594856177 p^{9} T^{4} + 5556 p^{18} T^{5} + p^{27} T^{6} \)
13$S_4\times C_2$ \( 1 + 83094 T + 20077033299 T^{2} + 754871996338436 T^{3} + 20077033299 p^{9} T^{4} + 83094 p^{18} T^{5} + p^{27} T^{6} \)
17$S_4\times C_2$ \( 1 - 367062 T + 268505189631 T^{2} - 69711192496917428 T^{3} + 268505189631 p^{9} T^{4} - 367062 p^{18} T^{5} + p^{27} T^{6} \)
19$S_4\times C_2$ \( 1 + 1489116 T + 1342580105481 T^{2} + 830560424045529832 T^{3} + 1342580105481 p^{9} T^{4} + 1489116 p^{18} T^{5} + p^{27} T^{6} \)
23$S_4\times C_2$ \( 1 - 499920 T - 191246545323 T^{2} + 4150866877865471776 T^{3} - 191246545323 p^{9} T^{4} - 499920 p^{18} T^{5} + p^{27} T^{6} \)
29$S_4\times C_2$ \( 1 - 5234682 T + 42736985914563 T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + 42736985914563 p^{9} T^{4} - 5234682 p^{18} T^{5} + p^{27} T^{6} \)
31$S_4\times C_2$ \( 1 + 12708912 T + 128130963574173 T^{2} + \)\(72\!\cdots\!04\)\( T^{3} + 128130963574173 p^{9} T^{4} + 12708912 p^{18} T^{5} + p^{27} T^{6} \)
37$S_4\times C_2$ \( 1 - 21724434 T + 286542561812475 T^{2} - \)\(28\!\cdots\!32\)\( T^{3} + 286542561812475 p^{9} T^{4} - 21724434 p^{18} T^{5} + p^{27} T^{6} \)
41$S_4\times C_2$ \( 1 - 27440478 T + 29653467030543 p T^{2} - \)\(18\!\cdots\!72\)\( T^{3} + 29653467030543 p^{10} T^{4} - 27440478 p^{18} T^{5} + p^{27} T^{6} \)
43$S_4\times C_2$ \( 1 - 23218260 T + 251755624778721 T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + 251755624778721 p^{9} T^{4} - 23218260 p^{18} T^{5} + p^{27} T^{6} \)
47$S_4\times C_2$ \( 1 - 28701528 T + 2689800471894429 T^{2} - \)\(45\!\cdots\!28\)\( T^{3} + 2689800471894429 p^{9} T^{4} - 28701528 p^{18} T^{5} + p^{27} T^{6} \)
53$S_4\times C_2$ \( 1 + 45629982 T + 7541283402476907 T^{2} + \)\(30\!\cdots\!96\)\( T^{3} + 7541283402476907 p^{9} T^{4} + 45629982 p^{18} T^{5} + p^{27} T^{6} \)
59$S_4\times C_2$ \( 1 - 268721868 T + 46457702853914817 T^{2} - \)\(50\!\cdots\!40\)\( T^{3} + 46457702853914817 p^{9} T^{4} - 268721868 p^{18} T^{5} + p^{27} T^{6} \)
61$S_4\times C_2$ \( 1 - 155970138 T + 27601780143557283 T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + 27601780143557283 p^{9} T^{4} - 155970138 p^{18} T^{5} + p^{27} T^{6} \)
67$S_4\times C_2$ \( 1 - 526916604 T + 169230786613422441 T^{2} - \)\(33\!\cdots\!48\)\( T^{3} + 169230786613422441 p^{9} T^{4} - 526916604 p^{18} T^{5} + p^{27} T^{6} \)
71$S_4\times C_2$ \( 1 - 239894424 T + 117827037562982037 T^{2} - \)\(16\!\cdots\!88\)\( T^{3} + 117827037562982037 p^{9} T^{4} - 239894424 p^{18} T^{5} + p^{27} T^{6} \)
73$S_4\times C_2$ \( 1 - 198362430 T + 177547240540777287 T^{2} - \)\(22\!\cdots\!56\)\( T^{3} + 177547240540777287 p^{9} T^{4} - 198362430 p^{18} T^{5} + p^{27} T^{6} \)
79$S_4\times C_2$ \( 1 - 413839728 T + 365148310414214253 T^{2} - \)\(92\!\cdots\!64\)\( T^{3} + 365148310414214253 p^{9} T^{4} - 413839728 p^{18} T^{5} + p^{27} T^{6} \)
83$S_4\times C_2$ \( 1 + 371949828 T + 338245334081122329 T^{2} + \)\(17\!\cdots\!88\)\( p T^{3} + 338245334081122329 p^{9} T^{4} + 371949828 p^{18} T^{5} + p^{27} T^{6} \)
89$S_4\times C_2$ \( 1 - 754926606 T + 926540806511526711 T^{2} - \)\(52\!\cdots\!12\)\( T^{3} + 926540806511526711 p^{9} T^{4} - 754926606 p^{18} T^{5} + p^{27} T^{6} \)
97$S_4\times C_2$ \( 1 + 903451002 T + 2439888940908067119 T^{2} + \)\(13\!\cdots\!72\)\( T^{3} + 2439888940908067119 p^{9} T^{4} + 903451002 p^{18} T^{5} + p^{27} T^{6} \)
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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

−11.09380559315768570319335514854, −10.77428236382325139607355177297, −10.23917861728535073896032419912, −10.10454035690325780307212076881, −9.536844592944351039172219365706, −9.245326409519962045678519061592, −8.919588234705491603995123870020, −8.217886097569461182435863282720, −8.081637652255168044024745732030, −7.61607118437349701582677039323, −7.04124496304808983719577702305, −6.47511628512369500305446761008, −6.39135352349608120122047453483, −5.73498748580020117360238736623, −5.34824589028862019333360751561, −5.18612424602002602278565491452, −4.63287395753991564806578506489, −3.86322046394280977812026842400, −3.84155883227607337060474503051, −2.56289362614638317153958130361, −2.33395494008679372245902473969, −2.16449764126106293444241531158, −1.39276211281363074806761488909, −0.76689926686129540612453069092, −0.48799360738727987325515356578, 0.48799360738727987325515356578, 0.76689926686129540612453069092, 1.39276211281363074806761488909, 2.16449764126106293444241531158, 2.33395494008679372245902473969, 2.56289362614638317153958130361, 3.84155883227607337060474503051, 3.86322046394280977812026842400, 4.63287395753991564806578506489, 5.18612424602002602278565491452, 5.34824589028862019333360751561, 5.73498748580020117360238736623, 6.39135352349608120122047453483, 6.47511628512369500305446761008, 7.04124496304808983719577702305, 7.61607118437349701582677039323, 8.081637652255168044024745732030, 8.217886097569461182435863282720, 8.919588234705491603995123870020, 9.245326409519962045678519061592, 9.536844592944351039172219365706, 10.10454035690325780307212076881, 10.23917861728535073896032419912, 10.77428236382325139607355177297, 11.09380559315768570319335514854

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