Dirichlet series
| L(s) = 1 | − 6·2-s + 9·3-s + 12·4-s − 10·5-s − 54·6-s + 14·7-s + 16·8-s + 27·9-s + 60·10-s − 88·11-s + 108·12-s + 9·13-s − 84·14-s − 90·15-s − 144·16-s + 84·17-s − 162·18-s + 32·19-s − 120·20-s + 126·21-s + 528·22-s + 2·23-s + 144·24-s + 279·25-s − 54·26-s − 54·27-s + 168·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3/2·4-s − 0.894·5-s − 3.67·6-s + 0.755·7-s + 0.707·8-s + 9-s + 1.89·10-s − 2.41·11-s + 2.59·12-s + 0.192·13-s − 1.60·14-s − 1.54·15-s − 9/4·16-s + 1.19·17-s − 2.12·18-s + 0.386·19-s − 1.34·20-s + 1.30·21-s + 5.11·22-s + 0.0181·23-s + 1.22·24-s + 2.23·25-s − 0.407·26-s − 0.384·27-s + 1.13·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 19^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(92603.1\) |
| Root analytic conductor: | \(2.59349\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.460924996\) |
| \(L(\frac12)\) | \(\approx\) | \(1.460924996\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| 3 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) | |
| 19 | \( 1 - 32 T - 1231 T^{2} + 54400 p T^{3} - 1231 p^{3} T^{4} - 32 p^{6} T^{5} + p^{9} T^{6} \) | |
| good | 5 | \( 1 + 2 p T - 179 T^{2} - 1202 T^{3} + 23006 T^{4} + 15634 T^{5} - 3402359 T^{6} + 15634 p^{3} T^{7} + 23006 p^{6} T^{8} - 1202 p^{9} T^{9} - 179 p^{12} T^{10} + 2 p^{16} T^{11} + p^{18} T^{12} \) |
| 7 | \( ( 1 - p T + 916 T^{2} - 629 p T^{3} + 916 p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} )^{2} \) | |
| 11 | \( ( 1 + 4 p T - 747 T^{2} - 100096 T^{3} - 747 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} )^{2} \) | |
| 13 | \( 1 - 9 T - 1881 T^{2} - 1072 p T^{3} - 477819 T^{4} + 30496689 T^{5} + 12364262070 T^{6} + 30496689 p^{3} T^{7} - 477819 p^{6} T^{8} - 1072 p^{10} T^{9} - 1881 p^{12} T^{10} - 9 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - 84 T + 141 p T^{2} - 382188 T^{3} - 1568442 T^{4} + 1796551404 T^{5} - 39760879367 T^{6} + 1796551404 p^{3} T^{7} - 1568442 p^{6} T^{8} - 382188 p^{9} T^{9} + 141 p^{13} T^{10} - 84 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 2 T - 6305 T^{2} + 298270 T^{3} - 37269010 T^{4} - 969721562 T^{5} + 3141587063179 T^{6} - 969721562 p^{3} T^{7} - 37269010 p^{6} T^{8} + 298270 p^{9} T^{9} - 6305 p^{12} T^{10} - 2 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 92 T - 34559 T^{2} - 7621708 T^{3} + 269018678 T^{4} + 110754147260 T^{5} + 12359272986757 T^{6} + 110754147260 p^{3} T^{7} + 269018678 p^{6} T^{8} - 7621708 p^{9} T^{9} - 34559 p^{12} T^{10} + 92 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( ( 1 + 109 T + 15884 T^{2} - 3227519 T^{3} + 15884 p^{3} T^{4} + 109 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 37 | \( ( 1 - 245 T + 80954 T^{2} - 26137769 T^{3} + 80954 p^{3} T^{4} - 245 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 41 | \( 1 - 688 T + 178021 T^{2} - 34071376 T^{3} + 8687586134 T^{4} - 5663148272 p T^{5} - 455041235673215 T^{6} - 5663148272 p^{4} T^{7} + 8687586134 p^{6} T^{8} - 34071376 p^{9} T^{9} + 178021 p^{12} T^{10} - 688 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 103 T - 148615 T^{2} + 28855470 T^{3} + 10362199195 T^{4} - 1646143554743 T^{5} - 565545995251946 T^{6} - 1646143554743 p^{3} T^{7} + 10362199195 p^{6} T^{8} + 28855470 p^{9} T^{9} - 148615 p^{12} T^{10} - 103 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 322 T - 41621 T^{2} - 110327606 T^{3} - 19268104366 T^{4} + 5170379504218 T^{5} + 5099902315885387 T^{6} + 5170379504218 p^{3} T^{7} - 19268104366 p^{6} T^{8} - 110327606 p^{9} T^{9} - 41621 p^{12} T^{10} + 322 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 1322 T + 767605 T^{2} - 374446142 T^{3} + 205718331302 T^{4} - 94945369502978 T^{5} + 37019491245693577 T^{6} - 94945369502978 p^{3} T^{7} + 205718331302 p^{6} T^{8} - 374446142 p^{9} T^{9} + 767605 p^{12} T^{10} - 1322 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 252 T - 563325 T^{2} - 48326292 T^{3} + 238184894850 T^{4} + 10399187473668 T^{5} - 55410445406069525 T^{6} + 10399187473668 p^{3} T^{7} + 238184894850 p^{6} T^{8} - 48326292 p^{9} T^{9} - 563325 p^{12} T^{10} + 252 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 435 T - 349161 T^{2} + 2632808 p T^{3} + 1340774865 p T^{4} - 6012252549 p^{2} T^{5} - 3543844324458 p^{2} T^{6} - 6012252549 p^{5} T^{7} + 1340774865 p^{7} T^{8} + 2632808 p^{10} T^{9} - 349161 p^{12} T^{10} - 435 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 719 T - 534855 T^{2} + 127967550 T^{3} + 472091302435 T^{4} - 1121674496117 p T^{5} - 28876561669226 p^{2} T^{6} - 1121674496117 p^{4} T^{7} + 472091302435 p^{6} T^{8} + 127967550 p^{9} T^{9} - 534855 p^{12} T^{10} - 719 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 62 T - 351677 T^{2} - 155369702 T^{3} + 3283454210 T^{4} + 31536492386842 T^{5} + 50410407058465987 T^{6} + 31536492386842 p^{3} T^{7} + 3283454210 p^{6} T^{8} - 155369702 p^{9} T^{9} - 351677 p^{12} T^{10} - 62 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 581 T - 319221 T^{2} + 663799836 T^{3} - 127822614479 T^{4} - 134331651269879 T^{5} + 172061413897952710 T^{6} - 134331651269879 p^{3} T^{7} - 127822614479 p^{6} T^{8} + 663799836 p^{9} T^{9} - 319221 p^{12} T^{10} - 581 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 489 T - 1145715 T^{2} + 183561926 T^{3} + 1024540014807 T^{4} - 47109257884197 T^{5} - 578738661463834122 T^{6} - 47109257884197 p^{3} T^{7} + 1024540014807 p^{6} T^{8} + 183561926 p^{9} T^{9} - 1145715 p^{12} T^{10} - 489 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( ( 1 - 2496 T + 3499089 T^{2} - 3226639296 T^{3} + 3499089 p^{3} T^{4} - 2496 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 89 | \( 1 + 1584 T + 321033 T^{2} - 209817792 T^{3} + 333844452834 T^{4} - 78185603887416 T^{5} - 543647357985948455 T^{6} - 78185603887416 p^{3} T^{7} + 333844452834 p^{6} T^{8} - 209817792 p^{9} T^{9} + 321033 p^{12} T^{10} + 1584 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 974 T - 2079643 T^{2} - 654591702 T^{3} + 4651618042378 T^{4} + 884600658264646 T^{5} - 4392104994237345107 T^{6} + 884600658264646 p^{3} T^{7} + 4651618042378 p^{6} T^{8} - 654591702 p^{9} T^{9} - 2079643 p^{12} T^{10} + 974 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−7.33979911940053856171184913567, −7.12506609299480865895281578604, −6.87305315639958888978941993333, −6.86938493680807970628693256970, −6.32425332911062646120108327208, −6.02897160987068082314794551288, −5.95217482979652277168261672740, −5.29695464713185838189864783887, −5.27806520592706187380906568668, −5.18329193318385472835001984799, −5.03318559707406621169362110875, −4.50155438833076379654555140690, −4.38350322644403943800679331087, −4.12468586059776912343222214405, −3.51707190987950777685456118786, −3.46670369489069848524575477029, −3.43950171241014554367180577565, −2.72742185328317744655246672136, −2.66156462675325810189319042848, −2.27718825712673483908890136924, −2.18951877015826184875813969342, −1.37244729643852202861215760880, −1.23161102649562794158986510908, −0.54822592171549979973705404020, −0.45759803224516073293198667891, 0.45759803224516073293198667891, 0.54822592171549979973705404020, 1.23161102649562794158986510908, 1.37244729643852202861215760880, 2.18951877015826184875813969342, 2.27718825712673483908890136924, 2.66156462675325810189319042848, 2.72742185328317744655246672136, 3.43950171241014554367180577565, 3.46670369489069848524575477029, 3.51707190987950777685456118786, 4.12468586059776912343222214405, 4.38350322644403943800679331087, 4.50155438833076379654555140690, 5.03318559707406621169362110875, 5.18329193318385472835001984799, 5.27806520592706187380906568668, 5.29695464713185838189864783887, 5.95217482979652277168261672740, 6.02897160987068082314794551288, 6.32425332911062646120108327208, 6.86938493680807970628693256970, 6.87305315639958888978941993333, 7.12506609299480865895281578604, 7.33979911940053856171184913567
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.