Dirichlet series
| L(s) = 1 | − 2-s − 18·3-s − 16·4-s + 30·5-s + 18·6-s + 23·7-s + 2·8-s + 189·9-s − 30·10-s − 111·11-s + 288·12-s − 83·13-s − 23·14-s − 540·15-s + 103·16-s − 35·17-s − 189·18-s − 76·19-s − 480·20-s − 414·21-s + 111·22-s + 166·23-s − 36·24-s + 525·25-s + 83·26-s − 1.51e3·27-s − 368·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 3.46·3-s − 2·4-s + 2.68·5-s + 1.22·6-s + 1.24·7-s + 0.0883·8-s + 7·9-s − 0.948·10-s − 3.04·11-s + 6.92·12-s − 1.77·13-s − 0.439·14-s − 9.29·15-s + 1.60·16-s − 0.499·17-s − 2.47·18-s − 0.917·19-s − 5.36·20-s − 4.30·21-s + 1.07·22-s + 1.50·23-s − 0.306·24-s + 21/5·25-s + 0.626·26-s − 10.7·27-s − 2.48·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{6} \cdot 5^{6} \cdot 29^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.85845\times 10^{8}\) |
| Root analytic conductor: | \(5.06614\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 3^{6} \cdot 5^{6} \cdot 29^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T )^{6} \) |
| 5 | \( ( 1 - p T )^{6} \) | |
| 29 | \( ( 1 + p T )^{6} \) | |
| good | 2 | \( 1 + T + 17 T^{2} + 31 T^{3} + 99 p T^{4} + 49 p^{3} T^{5} + 59 p^{5} T^{6} + 49 p^{6} T^{7} + 99 p^{7} T^{8} + 31 p^{9} T^{9} + 17 p^{12} T^{10} + p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 23 T + 254 p T^{2} - 33573 T^{3} + 200121 p T^{4} - 21133606 T^{5} + 89061764 p T^{6} - 21133606 p^{3} T^{7} + 200121 p^{7} T^{8} - 33573 p^{9} T^{9} + 254 p^{13} T^{10} - 23 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 111 T + 8518 T^{2} + 464997 T^{3} + 23569223 T^{4} + 1040397498 T^{5} + 41724498228 T^{6} + 1040397498 p^{3} T^{7} + 23569223 p^{6} T^{8} + 464997 p^{9} T^{9} + 8518 p^{12} T^{10} + 111 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 83 T + 10394 T^{2} + 723175 T^{3} + 50905879 T^{4} + 2854005298 T^{5} + 143794149004 T^{6} + 2854005298 p^{3} T^{7} + 50905879 p^{6} T^{8} + 723175 p^{9} T^{9} + 10394 p^{12} T^{10} + 83 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 35 T + 14404 T^{2} + 16755 T^{3} + 101023199 T^{4} - 1196999110 T^{5} + 569642423544 T^{6} - 1196999110 p^{3} T^{7} + 101023199 p^{6} T^{8} + 16755 p^{9} T^{9} + 14404 p^{12} T^{10} + 35 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 4 p T + 20286 T^{2} + 909940 T^{3} + 232082439 T^{4} + 9954125400 T^{5} + 1990504469860 T^{6} + 9954125400 p^{3} T^{7} + 232082439 p^{6} T^{8} + 909940 p^{9} T^{9} + 20286 p^{12} T^{10} + 4 p^{16} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 166 T + 51926 T^{2} - 7824746 T^{3} + 1364141423 T^{4} - 160814072852 T^{5} + 21405941738964 T^{6} - 160814072852 p^{3} T^{7} + 1364141423 p^{6} T^{8} - 7824746 p^{9} T^{9} + 51926 p^{12} T^{10} - 166 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 + 164 T + 3818 p T^{2} + 15948140 T^{3} + 6209022687 T^{4} + 730195824904 T^{5} + 213418250518548 T^{6} + 730195824904 p^{3} T^{7} + 6209022687 p^{6} T^{8} + 15948140 p^{9} T^{9} + 3818 p^{13} T^{10} + 164 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 538 T + 232170 T^{2} + 56187954 T^{3} + 9816484983 T^{4} + 801243597036 T^{5} + 52007584091404 T^{6} + 801243597036 p^{3} T^{7} + 9816484983 p^{6} T^{8} + 56187954 p^{9} T^{9} + 232170 p^{12} T^{10} + 538 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 1250 T + 977418 T^{2} + 539874226 T^{3} + 234862698639 T^{4} + 82066365274524 T^{5} + 23740188279796908 T^{6} + 82066365274524 p^{3} T^{7} + 234862698639 p^{6} T^{8} + 539874226 p^{9} T^{9} + 977418 p^{12} T^{10} + 1250 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 80 T + 307874 T^{2} + 7987248 T^{3} + 43904245863 T^{4} - 367000539488 T^{5} + 4096060870989308 T^{6} - 367000539488 p^{3} T^{7} + 43904245863 p^{6} T^{8} + 7987248 p^{9} T^{9} + 307874 p^{12} T^{10} + 80 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 491 T + 461586 T^{2} + 175924697 T^{3} + 102189500239 T^{4} + 31352686801646 T^{5} + 13470122252169724 T^{6} + 31352686801646 p^{3} T^{7} + 102189500239 p^{6} T^{8} + 175924697 p^{9} T^{9} + 461586 p^{12} T^{10} + 491 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 356 T + 371194 T^{2} + 46861380 T^{3} + 76020887111 T^{4} + 254860438120 p T^{5} + 16047353418765708 T^{6} + 254860438120 p^{4} T^{7} + 76020887111 p^{6} T^{8} + 46861380 p^{9} T^{9} + 371194 p^{12} T^{10} + 356 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 646 T + 230826 T^{2} + 121250906 T^{3} + 129528852519 T^{4} + 51938312437788 T^{5} + 19393269846300204 T^{6} + 51938312437788 p^{3} T^{7} + 129528852519 p^{6} T^{8} + 121250906 p^{9} T^{9} + 230826 p^{12} T^{10} + 646 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 1476 T + 1453298 T^{2} + 808430404 T^{3} + 305470051079 T^{4} + 39616783939272 T^{5} + 1350688756460924 T^{6} + 39616783939272 p^{3} T^{7} + 305470051079 p^{6} T^{8} + 808430404 p^{9} T^{9} + 1453298 p^{12} T^{10} + 1476 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 87 T + 906756 T^{2} + 90888947 T^{3} + 448258013543 T^{4} + 45110196087166 T^{5} + 168234343098240248 T^{6} + 45110196087166 p^{3} T^{7} + 448258013543 p^{6} T^{8} + 90888947 p^{9} T^{9} + 906756 p^{12} T^{10} - 87 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 1312 T + 1852810 T^{2} + 1611888992 T^{3} + 1495129896703 T^{4} + 985776399450944 T^{5} + 677312860674974860 T^{6} + 985776399450944 p^{3} T^{7} + 1495129896703 p^{6} T^{8} + 1611888992 p^{9} T^{9} + 1852810 p^{12} T^{10} + 1312 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 1142 T + 2493354 T^{2} + 2062833798 T^{3} + 2477819618127 T^{4} + 1542438786431316 T^{5} + 1291047832906314604 T^{6} + 1542438786431316 p^{3} T^{7} + 2477819618127 p^{6} T^{8} + 2062833798 p^{9} T^{9} + 2493354 p^{12} T^{10} + 1142 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 142 T + 2458734 T^{2} + 320729906 T^{3} + 2708345651167 T^{4} + 304363766362756 T^{5} + 1714109080591951268 T^{6} + 304363766362756 p^{3} T^{7} + 2708345651167 p^{6} T^{8} + 320729906 p^{9} T^{9} + 2458734 p^{12} T^{10} + 142 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 346 T + 2084290 T^{2} - 886869526 T^{3} + 2126163032775 T^{4} - 966403593855620 T^{5} + 1425044725386713884 T^{6} - 966403593855620 p^{3} T^{7} + 2126163032775 p^{6} T^{8} - 886869526 p^{9} T^{9} + 2084290 p^{12} T^{10} - 346 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 2827 T + 6275690 T^{2} + 9624456251 T^{3} + 12336995077727 T^{4} + 13012136105515922 T^{5} + 11822824409185958988 T^{6} + 13012136105515922 p^{3} T^{7} + 12336995077727 p^{6} T^{8} + 9624456251 p^{9} T^{9} + 6275690 p^{12} T^{10} + 2827 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 4 T + 3818566 T^{2} - 167389116 T^{3} + 6964456455615 T^{4} - 443125224740808 T^{5} + 7825662565329846388 T^{6} - 443125224740808 p^{3} T^{7} + 6964456455615 p^{6} T^{8} - 167389116 p^{9} T^{9} + 3818566 p^{12} T^{10} + 4 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−6.06769630753900304571383483871, −5.81041867043093180788462950149, −5.46920412329107535706262119468, −5.42158073114155094934730601884, −5.40066747585605701117937295123, −5.21425429754826160193572279118, −5.14836608064808602789628430411, −4.87389179771967450363832527028, −4.76780585349772531646707502313, −4.70913290699507212889234731374, −4.54909080291466974462386448262, −4.47359411889405413605184688862, −4.41346485463710854812546846409, −3.53456174271351236701878678208, −3.32631225932778033336399535631, −3.30012771818319658793766011066, −3.25471528702656562438546157741, −2.75442070496146557324554449374, −2.35819942251061732831962296145, −2.23134704414184882563806610373, −1.96849779262704836455072118971, −1.60882676411487435652880450978, −1.42622350228537665062605519054, −1.36958129817956984038366708115, −1.29275176410778924557727850677, 0, 0, 0, 0, 0, 0, 1.29275176410778924557727850677, 1.36958129817956984038366708115, 1.42622350228537665062605519054, 1.60882676411487435652880450978, 1.96849779262704836455072118971, 2.23134704414184882563806610373, 2.35819942251061732831962296145, 2.75442070496146557324554449374, 3.25471528702656562438546157741, 3.30012771818319658793766011066, 3.32631225932778033336399535631, 3.53456174271351236701878678208, 4.41346485463710854812546846409, 4.47359411889405413605184688862, 4.54909080291466974462386448262, 4.70913290699507212889234731374, 4.76780585349772531646707502313, 4.87389179771967450363832527028, 5.14836608064808602789628430411, 5.21425429754826160193572279118, 5.40066747585605701117937295123, 5.42158073114155094934730601884, 5.46920412329107535706262119468, 5.81041867043093180788462950149, 6.06769630753900304571383483871
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.