Dirichlet series
| L(s) = 1 | + 20·3-s + 25·5-s + 39·7-s − 376·9-s + 280·11-s + 1.21e3·13-s + 500·15-s + 796·17-s + 3.14e3·19-s + 780·21-s + 9.12e3·23-s − 5.82e3·25-s − 9.36e3·27-s + 1.30e4·29-s − 5.76e3·31-s + 5.60e3·33-s + 975·35-s + 2.16e4·37-s + 2.42e4·39-s + 3.22e3·41-s + 3.78e4·43-s − 9.40e3·45-s + 2.97e4·47-s − 3.22e4·49-s + 1.59e4·51-s − 9.97e3·53-s + 7.00e3·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s + 0.447·5-s + 0.300·7-s − 1.54·9-s + 0.697·11-s + 1.99·13-s + 0.573·15-s + 0.668·17-s + 1.99·19-s + 0.385·21-s + 3.59·23-s − 1.86·25-s − 2.47·27-s + 2.87·29-s − 1.07·31-s + 0.895·33-s + 0.134·35-s + 2.60·37-s + 2.55·39-s + 0.299·41-s + 3.12·43-s − 0.691·45-s + 1.96·47-s − 1.91·49-s + 0.857·51-s − 0.487·53-s + 0.312·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{12} \cdot 31^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.18718\times 10^{7}\) |
| Root analytic conductor: | \(4.45955\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{12} \cdot 31^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(25.44821837\) |
| \(L(\frac12)\) | \(\approx\) | \(25.44821837\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 31 | \( ( 1 + p^{2} T )^{6} \) | |
| good | 3 | \( 1 - 20 T + 776 T^{2} - 13672 T^{3} + 122669 p T^{4} - 1717676 p T^{5} + 3895984 p^{3} T^{6} - 1717676 p^{6} T^{7} + 122669 p^{11} T^{8} - 13672 p^{15} T^{9} + 776 p^{20} T^{10} - 20 p^{25} T^{11} + p^{30} T^{12} \) |
| 5 | \( 1 - p^{2} T + 6447 T^{2} - 26168 p T^{3} + 30352013 T^{4} - 148944867 p T^{5} + 104604962214 T^{6} - 148944867 p^{6} T^{7} + 30352013 p^{10} T^{8} - 26168 p^{16} T^{9} + 6447 p^{20} T^{10} - p^{27} T^{11} + p^{30} T^{12} \) | |
| 7 | \( 1 - 39 T + 33757 T^{2} + 99502 p T^{3} + 862256299 T^{4} + 2011025727 p T^{5} + 17277515180270 T^{6} + 2011025727 p^{6} T^{7} + 862256299 p^{10} T^{8} + 99502 p^{16} T^{9} + 33757 p^{20} T^{10} - 39 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 - 280 T + 607680 T^{2} - 179680724 T^{3} + 159220779831 T^{4} - 50651447453932 T^{5} + 28114843074942688 T^{6} - 50651447453932 p^{5} T^{7} + 159220779831 p^{10} T^{8} - 179680724 p^{15} T^{9} + 607680 p^{20} T^{10} - 280 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 - 1214 T + 1799836 T^{2} - 1420734086 T^{3} + 1272846319679 T^{4} - 805990962943540 T^{5} + 567010290988831064 T^{6} - 805990962943540 p^{5} T^{7} + 1272846319679 p^{10} T^{8} - 1420734086 p^{15} T^{9} + 1799836 p^{20} T^{10} - 1214 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 - 796 T + 5621450 T^{2} - 6475441668 T^{3} + 14198436430495 T^{4} - 19469657394975328 T^{5} + 23372885348048833516 T^{6} - 19469657394975328 p^{5} T^{7} + 14198436430495 p^{10} T^{8} - 6475441668 p^{15} T^{9} + 5621450 p^{20} T^{10} - 796 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 - 3147 T + 11946457 T^{2} - 28171071414 T^{3} + 66924116328679 T^{4} - 120835547720136103 T^{5} + \)\(21\!\cdots\!06\)\( T^{6} - 120835547720136103 p^{5} T^{7} + 66924116328679 p^{10} T^{8} - 28171071414 p^{15} T^{9} + 11946457 p^{20} T^{10} - 3147 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 - 9122 T + 63947726 T^{2} - 309545635078 T^{3} + 1248317212798767 T^{4} - 4061713971367048468 T^{5} + \)\(11\!\cdots\!64\)\( T^{6} - 4061713971367048468 p^{5} T^{7} + 1248317212798767 p^{10} T^{8} - 309545635078 p^{15} T^{9} + 63947726 p^{20} T^{10} - 9122 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 - 13020 T + 110114048 T^{2} - 679806165648 T^{3} + 4061487160760367 T^{4} - 22270713835604650932 T^{5} + \)\(11\!\cdots\!44\)\( T^{6} - 22270713835604650932 p^{5} T^{7} + 4061487160760367 p^{10} T^{8} - 679806165648 p^{15} T^{9} + 110114048 p^{20} T^{10} - 13020 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 - 21678 T + 479428708 T^{2} - 6251851735846 T^{3} + 81379187809657663 T^{4} - \)\(77\!\cdots\!80\)\( T^{5} + \)\(73\!\cdots\!72\)\( T^{6} - \)\(77\!\cdots\!80\)\( p^{5} T^{7} + 81379187809657663 p^{10} T^{8} - 6251851735846 p^{15} T^{9} + 479428708 p^{20} T^{10} - 21678 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 - 3227 T + 243793815 T^{2} - 434576652808 T^{3} + 31236426224638989 T^{4} - \)\(14\!\cdots\!53\)\( T^{5} + \)\(40\!\cdots\!26\)\( T^{6} - \)\(14\!\cdots\!53\)\( p^{5} T^{7} + 31236426224638989 p^{10} T^{8} - 434576652808 p^{15} T^{9} + 243793815 p^{20} T^{10} - 3227 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 - 37882 T + 1263446400 T^{2} - 27650574771422 T^{3} + 536389953699186527 T^{4} - \)\(80\!\cdots\!52\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} - \)\(80\!\cdots\!52\)\( p^{5} T^{7} + 536389953699186527 p^{10} T^{8} - 27650574771422 p^{15} T^{9} + 1263446400 p^{20} T^{10} - 37882 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 - 29708 T + 780424670 T^{2} - 18095300918884 T^{3} + 377960720783171679 T^{4} - \)\(65\!\cdots\!84\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} - \)\(65\!\cdots\!84\)\( p^{5} T^{7} + 377960720783171679 p^{10} T^{8} - 18095300918884 p^{15} T^{9} + 780424670 p^{20} T^{10} - 29708 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 + 9976 T + 1027324912 T^{2} + 842309717932 T^{3} + 581304305834059007 T^{4} - \)\(30\!\cdots\!32\)\( T^{5} + \)\(23\!\cdots\!48\)\( T^{6} - \)\(30\!\cdots\!32\)\( p^{5} T^{7} + 581304305834059007 p^{10} T^{8} + 842309717932 p^{15} T^{9} + 1027324912 p^{20} T^{10} + 9976 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 - 20573 T + 2532776797 T^{2} - 51247585442122 T^{3} + 3528351140722635539 T^{4} - \)\(61\!\cdots\!73\)\( T^{5} + \)\(31\!\cdots\!22\)\( T^{6} - \)\(61\!\cdots\!73\)\( p^{5} T^{7} + 3528351140722635539 p^{10} T^{8} - 51247585442122 p^{15} T^{9} + 2532776797 p^{20} T^{10} - 20573 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 - 21610 T + 2923138012 T^{2} - 59288705581866 T^{3} + 4849488202989226735 T^{4} - \)\(83\!\cdots\!44\)\( T^{5} + \)\(50\!\cdots\!24\)\( T^{6} - \)\(83\!\cdots\!44\)\( p^{5} T^{7} + 4849488202989226735 p^{10} T^{8} - 59288705581866 p^{15} T^{9} + 2923138012 p^{20} T^{10} - 21610 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 + 17024 T + 7631247778 T^{2} + 108165017026432 T^{3} + 24859918760903204263 T^{4} + \)\(28\!\cdots\!92\)\( T^{5} + \)\(44\!\cdots\!48\)\( T^{6} + \)\(28\!\cdots\!92\)\( p^{5} T^{7} + 24859918760903204263 p^{10} T^{8} + 108165017026432 p^{15} T^{9} + 7631247778 p^{20} T^{10} + 17024 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 - 44509 T + 7634253461 T^{2} - 236859785678878 T^{3} + 24009468582253800235 T^{4} - \)\(56\!\cdots\!29\)\( T^{5} + \)\(48\!\cdots\!50\)\( T^{6} - \)\(56\!\cdots\!29\)\( p^{5} T^{7} + 24009468582253800235 p^{10} T^{8} - 236859785678878 p^{15} T^{9} + 7634253461 p^{20} T^{10} - 44509 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 + 161864 T + 13829487050 T^{2} + 587537181499240 T^{3} + 1641220579404692991 T^{4} - \)\(15\!\cdots\!08\)\( T^{5} - \)\(10\!\cdots\!36\)\( T^{6} - \)\(15\!\cdots\!08\)\( p^{5} T^{7} + 1641220579404692991 p^{10} T^{8} + 587537181499240 p^{15} T^{9} + 13829487050 p^{20} T^{10} + 161864 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 + 24420 T + 7983205626 T^{2} + 4303913707996 p T^{3} + 31169112541736946831 T^{4} + \)\(18\!\cdots\!48\)\( T^{5} + \)\(99\!\cdots\!64\)\( T^{6} + \)\(18\!\cdots\!48\)\( p^{5} T^{7} + 31169112541736946831 p^{10} T^{8} + 4303913707996 p^{16} T^{9} + 7983205626 p^{20} T^{10} + 24420 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 + 114160 T + 21143335752 T^{2} + 1697541360274172 T^{3} + \)\(18\!\cdots\!95\)\( T^{4} + \)\(11\!\cdots\!36\)\( T^{5} + \)\(96\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!36\)\( p^{5} T^{7} + \)\(18\!\cdots\!95\)\( p^{10} T^{8} + 1697541360274172 p^{15} T^{9} + 21143335752 p^{20} T^{10} + 114160 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 + 199742 T + 33300169610 T^{2} + 3378484020675254 T^{3} + \)\(33\!\cdots\!75\)\( T^{4} + \)\(24\!\cdots\!28\)\( T^{5} + \)\(20\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!28\)\( p^{5} T^{7} + \)\(33\!\cdots\!75\)\( p^{10} T^{8} + 3378484020675254 p^{15} T^{9} + 33300169610 p^{20} T^{10} + 199742 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 + 282951 T + 56530847179 T^{2} + 8804366365525984 T^{3} + \)\(11\!\cdots\!65\)\( T^{4} + \)\(12\!\cdots\!93\)\( T^{5} + \)\(12\!\cdots\!34\)\( T^{6} + \)\(12\!\cdots\!93\)\( p^{5} T^{7} + \)\(11\!\cdots\!65\)\( p^{10} T^{8} + 8804366365525984 p^{15} T^{9} + 56530847179 p^{20} T^{10} + 282951 p^{25} T^{11} + p^{30} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.68382848207408357091728816174, −6.10648165391422828085434815360, −5.88188807769972636897994657511, −5.65449089839642605710919211951, −5.65110026217875176346876694008, −5.63452692542482317145926591207, −5.52583832192767422485251506890, −4.82481397143347998106309405764, −4.82383717444235248483058881595, −4.24171506067147643615828411530, −4.18171324569566739765782525161, −4.08687246693699246246038485823, −3.81081984581924638861940636832, −3.14982721652162649782968871568, −3.05484663371264101303303029245, −3.01764757730796250501303734012, −2.82659426090660513414638621643, −2.58554044321335663095507958696, −2.50135896657747747484831887840, −1.66444228502020080104937511009, −1.38054276073517071387131202870, −1.21778139780511622250447486670, −1.10050048098872588097638013723, −0.58808554757045799092166569084, −0.47167203008137704884443659948, 0.47167203008137704884443659948, 0.58808554757045799092166569084, 1.10050048098872588097638013723, 1.21778139780511622250447486670, 1.38054276073517071387131202870, 1.66444228502020080104937511009, 2.50135896657747747484831887840, 2.58554044321335663095507958696, 2.82659426090660513414638621643, 3.01764757730796250501303734012, 3.05484663371264101303303029245, 3.14982721652162649782968871568, 3.81081984581924638861940636832, 4.08687246693699246246038485823, 4.18171324569566739765782525161, 4.24171506067147643615828411530, 4.82383717444235248483058881595, 4.82481397143347998106309405764, 5.52583832192767422485251506890, 5.63452692542482317145926591207, 5.65110026217875176346876694008, 5.65449089839642605710919211951, 5.88188807769972636897994657511, 6.10648165391422828085434815360, 6.68382848207408357091728816174
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.