Dirichlet series
| L(s) = 1 | + 48·2-s − 82·3-s + 1.34e3·4-s + 750·5-s − 3.93e3·6-s − 1.94e3·7-s + 2.86e4·8-s − 2.74e3·9-s + 3.60e4·10-s − 442·11-s − 1.10e5·12-s − 4.01e3·13-s − 9.31e4·14-s − 6.15e4·15-s + 5.16e5·16-s − 3.71e4·17-s − 1.31e5·18-s − 6.81e4·19-s + 1.00e6·20-s + 1.59e5·21-s − 2.12e4·22-s − 7.17e4·23-s − 2.35e6·24-s + 3.28e5·25-s − 1.92e5·26-s + 3.88e5·27-s − 2.60e6·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s − 1.75·3-s + 21/2·4-s + 2.68·5-s − 7.43·6-s − 2.13·7-s + 19.7·8-s − 1.25·9-s + 11.3·10-s − 0.100·11-s − 18.4·12-s − 0.506·13-s − 9.06·14-s − 4.70·15-s + 63/2·16-s − 1.83·17-s − 5.32·18-s − 2.28·19-s + 28.1·20-s + 3.74·21-s − 0.424·22-s − 1.22·23-s − 34.7·24-s + 21/5·25-s − 2.14·26-s + 3.80·27-s − 22.4·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 5^{6} \cdot 31^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.24729\times 10^{11}\) |
| Root analytic conductor: | \(9.84069\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{6} \cdot 5^{6} \cdot 31^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T )^{6} \) |
| 5 | \( ( 1 - p^{3} T )^{6} \) | |
| 31 | \( ( 1 - p^{3} T )^{6} \) | |
| good | 3 | \( 1 + 82 T + 1052 p^{2} T^{2} + 68066 p^{2} T^{3} + 538916 p^{4} T^{4} + 26663440 p^{4} T^{5} + 493709848 p^{5} T^{6} + 26663440 p^{11} T^{7} + 538916 p^{18} T^{8} + 68066 p^{23} T^{9} + 1052 p^{30} T^{10} + 82 p^{35} T^{11} + p^{42} T^{12} \) |
| 7 | \( 1 + 1940 T + 5184982 T^{2} + 6690885528 T^{3} + 10436356906039 T^{4} + 10121290907558692 T^{5} + 1623112730247106172 p T^{6} + 10121290907558692 p^{7} T^{7} + 10436356906039 p^{14} T^{8} + 6690885528 p^{21} T^{9} + 5184982 p^{28} T^{10} + 1940 p^{35} T^{11} + p^{42} T^{12} \) | |
| 11 | \( 1 + 442 T + 102184216 T^{2} + 39236381950 T^{3} + 4602557652146599 T^{4} + 1462472801745122196 T^{5} + \)\(11\!\cdots\!48\)\( T^{6} + 1462472801745122196 p^{7} T^{7} + 4602557652146599 p^{14} T^{8} + 39236381950 p^{21} T^{9} + 102184216 p^{28} T^{10} + 442 p^{35} T^{11} + p^{42} T^{12} \) | |
| 13 | \( 1 + 4014 T + 173433488 T^{2} + 671295982430 T^{3} + 19735657518618207 T^{4} + 63177723604660883812 T^{5} + \)\(14\!\cdots\!52\)\( T^{6} + 63177723604660883812 p^{7} T^{7} + 19735657518618207 p^{14} T^{8} + 671295982430 p^{21} T^{9} + 173433488 p^{28} T^{10} + 4014 p^{35} T^{11} + p^{42} T^{12} \) | |
| 17 | \( 1 + 37182 T + 1627910292 T^{2} + 44685486060682 T^{3} + 1259619638464325222 T^{4} + \)\(29\!\cdots\!44\)\( T^{5} + \)\(64\!\cdots\!10\)\( T^{6} + \)\(29\!\cdots\!44\)\( p^{7} T^{7} + 1259619638464325222 p^{14} T^{8} + 44685486060682 p^{21} T^{9} + 1627910292 p^{28} T^{10} + 37182 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 + 68170 T + 3632034578 T^{2} + 85853517970968 T^{3} + 1818579137340822914 T^{4} - \)\(10\!\cdots\!46\)\( T^{5} - \)\(20\!\cdots\!90\)\( T^{6} - \)\(10\!\cdots\!46\)\( p^{7} T^{7} + 1818579137340822914 p^{14} T^{8} + 85853517970968 p^{21} T^{9} + 3632034578 p^{28} T^{10} + 68170 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 + 71730 T + 17929105874 T^{2} + 941553528290554 T^{3} + \)\(13\!\cdots\!95\)\( T^{4} + \)\(55\!\cdots\!44\)\( T^{5} + \)\(59\!\cdots\!44\)\( T^{6} + \)\(55\!\cdots\!44\)\( p^{7} T^{7} + \)\(13\!\cdots\!95\)\( p^{14} T^{8} + 941553528290554 p^{21} T^{9} + 17929105874 p^{28} T^{10} + 71730 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 + 300546 T + 100078585896 T^{2} + 20605093128211826 T^{3} + \)\(42\!\cdots\!75\)\( T^{4} + \)\(64\!\cdots\!92\)\( T^{5} + \)\(97\!\cdots\!12\)\( T^{6} + \)\(64\!\cdots\!92\)\( p^{7} T^{7} + \)\(42\!\cdots\!75\)\( p^{14} T^{8} + 20605093128211826 p^{21} T^{9} + 100078585896 p^{28} T^{10} + 300546 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 + 4866 T + 188204619874 T^{2} + 183697745352062 p T^{3} + \)\(14\!\cdots\!38\)\( T^{4} + \)\(46\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} + \)\(46\!\cdots\!20\)\( p^{7} T^{7} + \)\(14\!\cdots\!38\)\( p^{14} T^{8} + 183697745352062 p^{22} T^{9} + 188204619874 p^{28} T^{10} + 4866 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( 1 + 38280 T + 682395023448 T^{2} + 160052776816066088 T^{3} + \)\(19\!\cdots\!36\)\( T^{4} + \)\(83\!\cdots\!24\)\( T^{5} + \)\(40\!\cdots\!30\)\( T^{6} + \)\(83\!\cdots\!24\)\( p^{7} T^{7} + \)\(19\!\cdots\!36\)\( p^{14} T^{8} + 160052776816066088 p^{21} T^{9} + 682395023448 p^{28} T^{10} + 38280 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 + 289636 T + 930936981904 T^{2} + 167834334013322970 T^{3} + \)\(39\!\cdots\!16\)\( T^{4} + \)\(50\!\cdots\!74\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(50\!\cdots\!74\)\( p^{7} T^{7} + \)\(39\!\cdots\!16\)\( p^{14} T^{8} + 167834334013322970 p^{21} T^{9} + 930936981904 p^{28} T^{10} + 289636 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 + 1387980 T + 2554950972646 T^{2} + 2799888927860601524 T^{3} + \)\(30\!\cdots\!79\)\( T^{4} + \)\(24\!\cdots\!68\)\( T^{5} + \)\(43\!\cdots\!00\)\( p T^{6} + \)\(24\!\cdots\!68\)\( p^{7} T^{7} + \)\(30\!\cdots\!79\)\( p^{14} T^{8} + 2799888927860601524 p^{21} T^{9} + 2554950972646 p^{28} T^{10} + 1387980 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 + 2061800 T + 7336667379514 T^{2} + 10664517161281739134 T^{3} + \)\(40\!\cdots\!90\)\( p T^{4} + \)\(23\!\cdots\!66\)\( T^{5} + \)\(33\!\cdots\!04\)\( T^{6} + \)\(23\!\cdots\!66\)\( p^{7} T^{7} + \)\(40\!\cdots\!90\)\( p^{15} T^{8} + 10664517161281739134 p^{21} T^{9} + 7336667379514 p^{28} T^{10} + 2061800 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 + 5189426 T + 21629966023466 T^{2} + 61854237326873416628 T^{3} + \)\(15\!\cdots\!86\)\( T^{4} + \)\(30\!\cdots\!10\)\( T^{5} + \)\(52\!\cdots\!78\)\( T^{6} + \)\(30\!\cdots\!10\)\( p^{7} T^{7} + \)\(15\!\cdots\!86\)\( p^{14} T^{8} + 61854237326873416628 p^{21} T^{9} + 21629966023466 p^{28} T^{10} + 5189426 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 + 337160 T + 12034446956264 T^{2} + 15143538048964592328 T^{3} + \)\(56\!\cdots\!03\)\( T^{4} + \)\(12\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!36\)\( T^{6} + \)\(12\!\cdots\!92\)\( p^{7} T^{7} + \)\(56\!\cdots\!03\)\( p^{14} T^{8} + 15143538048964592328 p^{21} T^{9} + 12034446956264 p^{28} T^{10} + 337160 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 - 487628 T + 25011440917730 T^{2} - 29137809241979477512 T^{3} + \)\(27\!\cdots\!55\)\( T^{4} - \)\(42\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!20\)\( T^{6} - \)\(42\!\cdots\!68\)\( p^{7} T^{7} + \)\(27\!\cdots\!55\)\( p^{14} T^{8} - 29137809241979477512 p^{21} T^{9} + 25011440917730 p^{28} T^{10} - 487628 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( 1 + 4767670 T + 42083503473998 T^{2} + \)\(13\!\cdots\!28\)\( T^{3} + \)\(67\!\cdots\!50\)\( T^{4} + \)\(16\!\cdots\!98\)\( T^{5} + \)\(68\!\cdots\!26\)\( T^{6} + \)\(16\!\cdots\!98\)\( p^{7} T^{7} + \)\(67\!\cdots\!50\)\( p^{14} T^{8} + \)\(13\!\cdots\!28\)\( p^{21} T^{9} + 42083503473998 p^{28} T^{10} + 4767670 p^{35} T^{11} + p^{42} T^{12} \) | |
| 73 | \( 1 + 2337232 T + 45784655709740 T^{2} + 84897037923457994494 T^{3} + \)\(91\!\cdots\!62\)\( T^{4} + \)\(14\!\cdots\!34\)\( T^{5} + \)\(11\!\cdots\!54\)\( T^{6} + \)\(14\!\cdots\!34\)\( p^{7} T^{7} + \)\(91\!\cdots\!62\)\( p^{14} T^{8} + 84897037923457994494 p^{21} T^{9} + 45784655709740 p^{28} T^{10} + 2337232 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 - 537410 T + 90366889427070 T^{2} - 8747050074445574402 T^{3} + \)\(36\!\cdots\!95\)\( T^{4} + \)\(38\!\cdots\!68\)\( T^{5} + \)\(89\!\cdots\!84\)\( T^{6} + \)\(38\!\cdots\!68\)\( p^{7} T^{7} + \)\(36\!\cdots\!95\)\( p^{14} T^{8} - 8747050074445574402 p^{21} T^{9} + 90366889427070 p^{28} T^{10} - 537410 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 + 4608400 T + 70077956567860 T^{2} + 62324756135431100614 T^{3} + \)\(20\!\cdots\!56\)\( T^{4} - \)\(37\!\cdots\!86\)\( T^{5} + \)\(66\!\cdots\!68\)\( T^{6} - \)\(37\!\cdots\!86\)\( p^{7} T^{7} + \)\(20\!\cdots\!56\)\( p^{14} T^{8} + 62324756135431100614 p^{21} T^{9} + 70077956567860 p^{28} T^{10} + 4608400 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( 1 + 4507442 T + 158715051419666 T^{2} + \)\(56\!\cdots\!94\)\( T^{3} + \)\(12\!\cdots\!71\)\( T^{4} + \)\(42\!\cdots\!08\)\( T^{5} + \)\(69\!\cdots\!68\)\( T^{6} + \)\(42\!\cdots\!08\)\( p^{7} T^{7} + \)\(12\!\cdots\!71\)\( p^{14} T^{8} + \)\(56\!\cdots\!94\)\( p^{21} T^{9} + 158715051419666 p^{28} T^{10} + 4507442 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( 1 - 1144396 T + 90454082946870 T^{2} - \)\(61\!\cdots\!52\)\( T^{3} + \)\(16\!\cdots\!39\)\( T^{4} - \)\(61\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} - \)\(61\!\cdots\!20\)\( p^{7} T^{7} + \)\(16\!\cdots\!39\)\( p^{14} T^{8} - \)\(61\!\cdots\!52\)\( p^{21} T^{9} + 90454082946870 p^{28} T^{10} - 1144396 p^{35} T^{11} + p^{42} T^{12} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.91486339636051344144880325431, −5.36715484660632041665215200680, −5.14048062919175964589506829393, −5.10741920882688149629593518235, −5.01759672551324242755799570002, −4.94597719147456226751517836428, −4.79098768006532100946428626318, −4.44393313275671922441235198630, −4.09808038075803299390134656453, −4.04253059499417016787616349193, −3.74489024643193321780371409673, −3.69197962115005320467192220110, −3.62562459380153275215324550419, −2.91024368018288788483939381697, −2.85893757950931826261771522621, −2.84190602031654606988505176634, −2.64341041570124475430228411471, −2.62293565248913099989068590007, −2.60612028579311614054553112288, −2.02411998328923736444497848078, −1.68888814389620601195689463523, −1.68502534096598989308770865996, −1.60150677713089439128581603557, −1.32915578380462555933040437648, −1.17840449398538609382172540046, 0, 0, 0, 0, 0, 0, 1.17840449398538609382172540046, 1.32915578380462555933040437648, 1.60150677713089439128581603557, 1.68502534096598989308770865996, 1.68888814389620601195689463523, 2.02411998328923736444497848078, 2.60612028579311614054553112288, 2.62293565248913099989068590007, 2.64341041570124475430228411471, 2.84190602031654606988505176634, 2.85893757950931826261771522621, 2.91024368018288788483939381697, 3.62562459380153275215324550419, 3.69197962115005320467192220110, 3.74489024643193321780371409673, 4.04253059499417016787616349193, 4.09808038075803299390134656453, 4.44393313275671922441235198630, 4.79098768006532100946428626318, 4.94597719147456226751517836428, 5.01759672551324242755799570002, 5.10741920882688149629593518235, 5.14048062919175964589506829393, 5.36715484660632041665215200680, 5.91486339636051344144880325431
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.