Dirichlet series
| L(s) = 1 | + 2·5-s − 42·7-s − 51·9-s + 66·11-s − 6·13-s − 14·17-s − 80·19-s − 254·23-s − 263·25-s − 30·27-s + 132·29-s + 52·31-s − 84·35-s − 518·37-s + 486·41-s − 428·43-s − 102·45-s − 790·47-s + 1.02e3·49-s − 40·53-s + 132·55-s − 436·59-s + 1.03e3·61-s + 2.14e3·63-s − 12·65-s − 562·67-s − 2.47e3·71-s + ⋯ |
| L(s) = 1 | + 0.178·5-s − 2.26·7-s − 1.88·9-s + 1.80·11-s − 0.128·13-s − 0.199·17-s − 0.965·19-s − 2.30·23-s − 2.10·25-s − 0.213·27-s + 0.845·29-s + 0.301·31-s − 0.405·35-s − 2.30·37-s + 1.85·41-s − 1.51·43-s − 0.337·45-s − 2.45·47-s + 3·49-s − 0.103·53-s + 0.323·55-s − 0.962·59-s + 2.17·61-s + 4.28·63-s − 0.0228·65-s − 1.02·67-s − 4.13·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 7^{6} \cdot 11^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.47523\times 10^{11}\) |
| Root analytic conductor: | \(8.52586\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{24} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
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 | 2 | \( 1 \) |
| 7 | \( ( 1 + p T )^{6} \) | |
| 11 | \( ( 1 - p T )^{6} \) | |
| good | 3 | \( 1 + 17 p T^{2} + 10 p T^{3} + 521 p T^{4} + 818 p T^{5} + 33458 T^{6} + 818 p^{4} T^{7} + 521 p^{7} T^{8} + 10 p^{10} T^{9} + 17 p^{13} T^{10} + p^{18} T^{12} \) |
| 5 | \( 1 - 2 T + 267 T^{2} + 78 T^{3} + 28059 T^{4} - 30628 T^{5} + 2913874 T^{6} - 30628 p^{3} T^{7} + 28059 p^{6} T^{8} + 78 p^{9} T^{9} + 267 p^{12} T^{10} - 2 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 6 T + 7226 T^{2} - 12986 T^{3} + 21595335 T^{4} - 212545140 T^{5} + 46341531436 T^{6} - 212545140 p^{3} T^{7} + 21595335 p^{6} T^{8} - 12986 p^{9} T^{9} + 7226 p^{12} T^{10} + 6 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 14 T + 13486 T^{2} + 38686 p T^{3} + 90635967 T^{4} + 7350497812 T^{5} + 447812918500 T^{6} + 7350497812 p^{3} T^{7} + 90635967 p^{6} T^{8} + 38686 p^{10} T^{9} + 13486 p^{12} T^{10} + 14 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 80 T + 20430 T^{2} + 808800 T^{3} + 107899847 T^{4} - 1308862768 T^{5} + 219419401156 T^{6} - 1308862768 p^{3} T^{7} + 107899847 p^{6} T^{8} + 808800 p^{9} T^{9} + 20430 p^{12} T^{10} + 80 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 + 254 T + 70315 T^{2} + 12243950 T^{3} + 2051483067 T^{4} + 263880838960 T^{5} + 32746801812082 T^{6} + 263880838960 p^{3} T^{7} + 2051483067 p^{6} T^{8} + 12243950 p^{9} T^{9} + 70315 p^{12} T^{10} + 254 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 - 132 T + 99898 T^{2} - 9991364 T^{3} + 4823385367 T^{4} - 405808479304 T^{5} + 147113503501772 T^{6} - 405808479304 p^{3} T^{7} + 4823385367 p^{6} T^{8} - 9991364 p^{9} T^{9} + 99898 p^{12} T^{10} - 132 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 52 T + 89883 T^{2} + 2087062 T^{3} + 3252254323 T^{4} + 377234281626 T^{5} + 87166967790482 T^{6} + 377234281626 p^{3} T^{7} + 3252254323 p^{6} T^{8} + 2087062 p^{9} T^{9} + 89883 p^{12} T^{10} - 52 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 14 p T + 269207 T^{2} + 85165306 T^{3} + 28898877291 T^{4} + 7179634256120 T^{5} + 1864618903150618 T^{6} + 7179634256120 p^{3} T^{7} + 28898877291 p^{6} T^{8} + 85165306 p^{9} T^{9} + 269207 p^{12} T^{10} + 14 p^{16} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 486 T + 323230 T^{2} - 119780726 T^{3} + 46993119439 T^{4} - 14376699282932 T^{5} + 4122837682268676 T^{6} - 14376699282932 p^{3} T^{7} + 46993119439 p^{6} T^{8} - 119780726 p^{9} T^{9} + 323230 p^{12} T^{10} - 486 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 428 T + 393842 T^{2} + 133211668 T^{3} + 71876415271 T^{4} + 19044270193336 T^{5} + 7379742977723868 T^{6} + 19044270193336 p^{3} T^{7} + 71876415271 p^{6} T^{8} + 133211668 p^{9} T^{9} + 393842 p^{12} T^{10} + 428 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 790 T + 642258 T^{2} + 333678938 T^{3} + 3492337457 p T^{4} + 62727363087012 T^{5} + 22537508139391772 T^{6} + 62727363087012 p^{3} T^{7} + 3492337457 p^{7} T^{8} + 333678938 p^{9} T^{9} + 642258 p^{12} T^{10} + 790 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 40 T + 208850 T^{2} - 40763096 T^{3} + 24540166119 T^{4} - 1549444871536 T^{5} + 5015035290868764 T^{6} - 1549444871536 p^{3} T^{7} + 24540166119 p^{6} T^{8} - 40763096 p^{9} T^{9} + 208850 p^{12} T^{10} + 40 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 436 T + 945043 T^{2} + 344755578 T^{3} + 414850671851 T^{4} + 125541076038526 T^{5} + 108238520466107794 T^{6} + 125541076038526 p^{3} T^{7} + 414850671851 p^{6} T^{8} + 344755578 p^{9} T^{9} + 945043 p^{12} T^{10} + 436 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 1034 T + 1213638 T^{2} - 827989170 T^{3} + 607908506471 T^{4} - 325318914419660 T^{5} + 178537931833608916 T^{6} - 325318914419660 p^{3} T^{7} + 607908506471 p^{6} T^{8} - 827989170 p^{9} T^{9} + 1213638 p^{12} T^{10} - 1034 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 562 T + 1314091 T^{2} + 416909682 T^{3} + 681384395571 T^{4} + 123413911558312 T^{5} + 227606452205865378 T^{6} + 123413911558312 p^{3} T^{7} + 681384395571 p^{6} T^{8} + 416909682 p^{9} T^{9} + 1314091 p^{12} T^{10} + 562 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 2474 T + 4156019 T^{2} + 4921377706 T^{3} + 4708740035227 T^{4} + 3654320624597408 T^{5} + 474943847471938 p^{2} T^{6} + 3654320624597408 p^{3} T^{7} + 4708740035227 p^{6} T^{8} + 4921377706 p^{9} T^{9} + 4156019 p^{12} T^{10} + 2474 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 902 T + 1464166 T^{2} + 1154693750 T^{3} + 1041995682511 T^{4} + 711491269010964 T^{5} + 487603872205165556 T^{6} + 711491269010964 p^{3} T^{7} + 1041995682511 p^{6} T^{8} + 1154693750 p^{9} T^{9} + 1464166 p^{12} T^{10} + 902 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 1636 T + 2498302 T^{2} + 33331924 p T^{3} + 2469987735583 T^{4} + 1899577945348872 T^{5} + 1461465898798234628 T^{6} + 1899577945348872 p^{3} T^{7} + 2469987735583 p^{6} T^{8} + 33331924 p^{10} T^{9} + 2498302 p^{12} T^{10} + 1636 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 + 3016 T + 6643466 T^{2} + 10104768056 T^{3} + 12515177745655 T^{4} + 12402952946307472 T^{5} + 10338749994204949036 T^{6} + 12402952946307472 p^{3} T^{7} + 12515177745655 p^{6} T^{8} + 10104768056 p^{9} T^{9} + 6643466 p^{12} T^{10} + 3016 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 - 1750 T + 4497595 T^{2} - 5583770934 T^{3} + 8155725959611 T^{4} - 7522990771478308 T^{5} + 7722629637376363346 T^{6} - 7522990771478308 p^{3} T^{7} + 8155725959611 p^{6} T^{8} - 5583770934 p^{9} T^{9} + 4497595 p^{12} T^{10} - 1750 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 1250 T + 2518379 T^{2} + 1593791166 T^{3} + 1571798248867 T^{4} - 304189816010912 T^{5} + 214623295214286178 T^{6} - 304189816010912 p^{3} T^{7} + 1571798248867 p^{6} T^{8} + 1593791166 p^{9} T^{9} + 2518379 p^{12} T^{10} + 1250 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−5.33504046785270954536945638056, −5.08314460241253563282372964997, −4.71145446404979194842971755679, −4.57492196384778921571260815736, −4.56621241759995096655779570342, −4.30678762169290093554357319575, −4.24571104849646553433490848061, −4.06521302041563926912556724862, −3.71027064984486985497784032491, −3.70009535075218910414393666736, −3.56554647511755633936860602897, −3.42600235738734403691786764798, −3.42538556704132538781174499830, −2.86655449367963239661397147184, −2.86478931064933179518261142187, −2.72397453781365153282787214351, −2.50227899117176488301048214467, −2.39947265662062544018605350874, −2.34064403223413215077461014651, −1.71170853773646993767696594174, −1.65508884688342643867650367389, −1.51460663808947003980653283875, −1.45131547181095145990516625340, −1.08742501638657976948156490682, −0.918259867719945211447512254193, 0, 0, 0, 0, 0, 0, 0.918259867719945211447512254193, 1.08742501638657976948156490682, 1.45131547181095145990516625340, 1.51460663808947003980653283875, 1.65508884688342643867650367389, 1.71170853773646993767696594174, 2.34064403223413215077461014651, 2.39947265662062544018605350874, 2.50227899117176488301048214467, 2.72397453781365153282787214351, 2.86478931064933179518261142187, 2.86655449367963239661397147184, 3.42538556704132538781174499830, 3.42600235738734403691786764798, 3.56554647511755633936860602897, 3.70009535075218910414393666736, 3.71027064984486985497784032491, 4.06521302041563926912556724862, 4.24571104849646553433490848061, 4.30678762169290093554357319575, 4.56621241759995096655779570342, 4.57492196384778921571260815736, 4.71145446404979194842971755679, 5.08314460241253563282372964997, 5.33504046785270954536945638056
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.