Dirichlet series
| L(s) = 1 | − 4·2-s + 4-s − 30·5-s + 40·7-s + 6·8-s + 120·10-s − 88·11-s + 20·13-s − 160·14-s − 29·16-s − 124·17-s − 46·19-s − 30·20-s + 352·22-s − 210·23-s + 525·25-s − 80·26-s + 40·28-s − 296·29-s − 104·31-s + 132·32-s + 496·34-s − 1.20e3·35-s − 204·37-s + 184·38-s − 180·40-s − 344·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/8·4-s − 2.68·5-s + 2.15·7-s + 0.265·8-s + 3.79·10-s − 2.41·11-s + 0.426·13-s − 3.05·14-s − 0.453·16-s − 1.76·17-s − 0.555·19-s − 0.335·20-s + 3.41·22-s − 1.90·23-s + 21/5·25-s − 0.603·26-s + 0.269·28-s − 1.89·29-s − 0.602·31-s + 0.729·32-s + 2.50·34-s − 5.79·35-s − 0.906·37-s + 0.785·38-s − 0.711·40-s − 1.31·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{24} \cdot 5^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.86177\times 10^{8}\) |
| Root analytic conductor: | \(4.88833\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 3^{24} \cdot 5^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
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 | \( 1 \) |
| 5 | \( ( 1 + p T )^{6} \) | |
| good | 2 | \( 1 + p^{2} T + 15 T^{2} + 25 p T^{3} + 95 p T^{4} + 151 p^{2} T^{5} + 509 p^{2} T^{6} + 151 p^{5} T^{7} + 95 p^{7} T^{8} + 25 p^{10} T^{9} + 15 p^{12} T^{10} + p^{17} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 40 T + 1795 T^{2} - 7100 p T^{3} + 1354639 T^{4} - 28754356 T^{5} + 585312734 T^{6} - 28754356 p^{3} T^{7} + 1354639 p^{6} T^{8} - 7100 p^{10} T^{9} + 1795 p^{12} T^{10} - 40 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 8 p T + 864 p T^{2} + 581120 T^{3} + 34487488 T^{4} + 1540498408 T^{5} + 5713541278 p T^{6} + 1540498408 p^{3} T^{7} + 34487488 p^{6} T^{8} + 581120 p^{9} T^{9} + 864 p^{13} T^{10} + 8 p^{16} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 20 T + 8663 T^{2} - 126228 T^{3} + 34744651 T^{4} - 360615208 T^{5} + 90061908778 T^{6} - 360615208 p^{3} T^{7} + 34744651 p^{6} T^{8} - 126228 p^{9} T^{9} + 8663 p^{12} T^{10} - 20 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 124 T + 24474 T^{2} + 134308 p T^{3} + 259934659 T^{4} + 18722277280 T^{5} + 1609913951204 T^{6} + 18722277280 p^{3} T^{7} + 259934659 p^{6} T^{8} + 134308 p^{10} T^{9} + 24474 p^{12} T^{10} + 124 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 46 T + 22985 T^{2} + 705414 T^{3} + 253542922 T^{4} + 5221909598 T^{5} + 1923956438809 T^{6} + 5221909598 p^{3} T^{7} + 253542922 p^{6} T^{8} + 705414 p^{9} T^{9} + 22985 p^{12} T^{10} + 46 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 + 210 T + 47223 T^{2} + 7466790 T^{3} + 1101124035 T^{4} + 147004381380 T^{5} + 16748229860890 T^{6} + 147004381380 p^{3} T^{7} + 1101124035 p^{6} T^{8} + 7466790 p^{9} T^{9} + 47223 p^{12} T^{10} + 210 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 296 T + 156960 T^{2} + 32819824 T^{3} + 9752776288 T^{4} + 1530082088552 T^{5} + 318643723898570 T^{6} + 1530082088552 p^{3} T^{7} + 9752776288 p^{6} T^{8} + 32819824 p^{9} T^{9} + 156960 p^{12} T^{10} + 296 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 + 104 T + 120472 T^{2} + 14112508 T^{3} + 6877930888 T^{4} + 798524069648 T^{5} + 248771575721078 T^{6} + 798524069648 p^{3} T^{7} + 6877930888 p^{6} T^{8} + 14112508 p^{9} T^{9} + 120472 p^{12} T^{10} + 104 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 204 T + 229518 T^{2} + 31409012 T^{3} + 23106132267 T^{4} + 2366964443472 T^{5} + 1434960824277324 T^{6} + 2366964443472 p^{3} T^{7} + 23106132267 p^{6} T^{8} + 31409012 p^{9} T^{9} + 229518 p^{12} T^{10} + 204 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 344 T + 228333 T^{2} + 70339720 T^{3} + 30507613174 T^{4} + 7601643608024 T^{5} + 2604569378220521 T^{6} + 7601643608024 p^{3} T^{7} + 30507613174 p^{6} T^{8} + 70339720 p^{9} T^{9} + 228333 p^{12} T^{10} + 344 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 512 T + 304490 T^{2} - 114035784 T^{3} + 48130647691 T^{4} - 14537104221784 T^{5} + 4628583019035940 T^{6} - 14537104221784 p^{3} T^{7} + 48130647691 p^{6} T^{8} - 114035784 p^{9} T^{9} + 304490 p^{12} T^{10} - 512 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 238 T + 195015 T^{2} + 36986414 T^{3} + 7407783391 T^{4} + 4571239328 p T^{5} - 477211568990650 T^{6} + 4571239328 p^{4} T^{7} + 7407783391 p^{6} T^{8} + 36986414 p^{9} T^{9} + 195015 p^{12} T^{10} + 238 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 850 T + 343107 T^{2} + 31107110 T^{3} - 2956444769 T^{4} + 6953131975360 T^{5} + 6251628297772334 T^{6} + 6953131975360 p^{3} T^{7} - 2956444769 p^{6} T^{8} + 31107110 p^{9} T^{9} + 343107 p^{12} T^{10} + 850 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 1840 T + 2531721 T^{2} + 2321469680 T^{3} + 1764537195478 T^{4} + 1044786986053360 T^{5} + 528376754967850397 T^{6} + 1044786986053360 p^{3} T^{7} + 1764537195478 p^{6} T^{8} + 2321469680 p^{9} T^{9} + 2531721 p^{12} T^{10} + 1840 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 364 T + 347798 T^{2} + 94535052 T^{3} + 150582316951 T^{4} + 40551217923320 T^{5} + 35591274035032852 T^{6} + 40551217923320 p^{3} T^{7} + 150582316951 p^{6} T^{8} + 94535052 p^{9} T^{9} + 347798 p^{12} T^{10} + 364 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 88 T + 490906 T^{2} + 156966904 T^{3} + 159189276583 T^{4} + 9632231390576 T^{5} + 80853334627212524 T^{6} + 9632231390576 p^{3} T^{7} + 159189276583 p^{6} T^{8} + 156966904 p^{9} T^{9} + 490906 p^{12} T^{10} - 88 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 1364 T + 1151472 T^{2} + 629760820 T^{3} + 269997702880 T^{4} + 18102353174084 T^{5} - 21873842560191886 T^{6} + 18102353174084 p^{3} T^{7} + 269997702880 p^{6} T^{8} + 629760820 p^{9} T^{9} + 1151472 p^{12} T^{10} + 1364 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 836 T + 1203194 T^{2} - 845045868 T^{3} + 764405490259 T^{4} - 417329100597904 T^{5} + 337538834495072164 T^{6} - 417329100597904 p^{3} T^{7} + 764405490259 p^{6} T^{8} - 845045868 p^{9} T^{9} + 1203194 p^{12} T^{10} - 836 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 680 T + 1330438 T^{2} + 586068856 T^{3} + 849797963887 T^{4} + 287191056360272 T^{5} + 414896870576612756 T^{6} + 287191056360272 p^{3} T^{7} + 849797963887 p^{6} T^{8} + 586068856 p^{9} T^{9} + 1330438 p^{12} T^{10} + 680 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 + 2148 T + 4085814 T^{2} + 5292561012 T^{3} + 5937736769547 T^{4} + 5536580714120544 T^{5} + 4496650388666753884 T^{6} + 5536580714120544 p^{3} T^{7} + 5937736769547 p^{6} T^{8} + 5292561012 p^{9} T^{9} + 4085814 p^{12} T^{10} + 2148 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 3000 T + 6331152 T^{2} + 9569726760 T^{3} + 12126222621024 T^{4} + 12808596417843000 T^{5} + 11672528635101585634 T^{6} + 12808596417843000 p^{3} T^{7} + 12126222621024 p^{6} T^{8} + 9569726760 p^{9} T^{9} + 6331152 p^{12} T^{10} + 3000 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 612 T + 2308326 T^{2} + 1247419244 T^{3} + 4111793218419 T^{4} + 1842237308085648 T^{5} + 4200366189151667676 T^{6} + 1842237308085648 p^{3} T^{7} + 4111793218419 p^{6} T^{8} + 1247419244 p^{9} T^{9} + 2308326 p^{12} T^{10} + 612 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−6.29566170091375281375861737049, −5.89826146507742908754473171440, −5.79637476942155064381515005741, −5.52990790476410487181551508736, −5.50623588266367123197240110177, −5.19108258179530469870889758370, −4.97276912801519591427249127120, −4.81717140566585197198405800835, −4.54369634725585768599320562973, −4.45330665342690989327909899431, −4.36798851081485363189753652221, −4.30621252219043402673230627731, −4.16866446237938234598204489329, −3.60635695483203289651526544597, −3.45305300768411989762820834204, −3.36353748249892561510787155764, −3.24896545210859475129222194361, −2.79559503930661580151751817198, −2.47430479778886850575756486773, −2.35426089042293201409759156858, −2.28182641632147444900880640057, −1.61137702995729472119416865259, −1.47629550824667925897016393317, −1.36430257258629478525139487480, −1.31545737817216636993057826638, 0, 0, 0, 0, 0, 0, 1.31545737817216636993057826638, 1.36430257258629478525139487480, 1.47629550824667925897016393317, 1.61137702995729472119416865259, 2.28182641632147444900880640057, 2.35426089042293201409759156858, 2.47430479778886850575756486773, 2.79559503930661580151751817198, 3.24896545210859475129222194361, 3.36353748249892561510787155764, 3.45305300768411989762820834204, 3.60635695483203289651526544597, 4.16866446237938234598204489329, 4.30621252219043402673230627731, 4.36798851081485363189753652221, 4.45330665342690989327909899431, 4.54369634725585768599320562973, 4.81717140566585197198405800835, 4.97276912801519591427249127120, 5.19108258179530469870889758370, 5.50623588266367123197240110177, 5.52990790476410487181551508736, 5.79637476942155064381515005741, 5.89826146507742908754473171440, 6.29566170091375281375861737049
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.