Dirichlet series
| L(s) = 1 | + 3·3-s + 6·4-s + 6·9-s − 12·11-s + 18·12-s + 6·13-s + 17·16-s − 12·17-s − 3·19-s − 15·23-s + 16·25-s + 18·27-s − 15·29-s − 36·33-s + 36·36-s + 6·37-s + 18·39-s − 9·41-s + 3·43-s − 72·44-s − 30·47-s + 51·48-s − 36·51-s + 36·52-s + 9·53-s − 9·57-s + 36·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 3·4-s + 2·9-s − 3.61·11-s + 5.19·12-s + 1.66·13-s + 17/4·16-s − 2.91·17-s − 0.688·19-s − 3.12·23-s + 16/5·25-s + 3.46·27-s − 2.78·29-s − 6.26·33-s + 6·36-s + 0.986·37-s + 2.88·39-s − 1.40·41-s + 0.457·43-s − 10.8·44-s − 4.37·47-s + 7.36·48-s − 5.04·51-s + 4.99·52-s + 1.23·53-s − 1.19·57-s + 4.68·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(293195.\) |
| Root analytic conductor: | \(1.87654\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 7^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.458928621\) |
| \(L(\frac12)\) | \(\approx\) | \(4.458928621\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - p T + p T^{2} - p^{2} T^{3} + 7 p T^{4} - p^{3} T^{5} + 7 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} - p^{5} T^{9} + p^{5} T^{10} \) |
| 7 | \( 1 \) | |
| good | 2 | \( 1 - 3 p T^{2} + 19 T^{4} - 47 T^{6} + 103 T^{8} - 209 T^{10} + 103 p^{2} T^{12} - 47 p^{4} T^{14} + 19 p^{6} T^{16} - 3 p^{9} T^{18} + p^{10} T^{20} \) |
| 5 | \( 1 - 16 T^{2} - 12 T^{3} + 129 T^{4} + 147 T^{5} - 726 T^{6} - 144 p T^{7} + 3663 T^{8} + 1257 T^{9} - 18069 T^{10} + 1257 p T^{11} + 3663 p^{2} T^{12} - 144 p^{4} T^{13} - 726 p^{4} T^{14} + 147 p^{5} T^{15} + 129 p^{6} T^{16} - 12 p^{7} T^{17} - 16 p^{8} T^{18} + p^{10} T^{20} \) | |
| 11 | \( 1 + 12 T + 105 T^{2} + 684 T^{3} + 3745 T^{4} + 17382 T^{5} + 72307 T^{6} + 270561 T^{7} + 952558 T^{8} + 3194589 T^{9} + 10673917 T^{10} + 3194589 p T^{11} + 952558 p^{2} T^{12} + 270561 p^{3} T^{13} + 72307 p^{4} T^{14} + 17382 p^{5} T^{15} + 3745 p^{6} T^{16} + 684 p^{7} T^{17} + 105 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 6 T + 68 T^{2} - 336 T^{3} + 2292 T^{4} - 723 p T^{5} + 51837 T^{6} - 187401 T^{7} + 909867 T^{8} - 3004662 T^{9} + 13054461 T^{10} - 3004662 p T^{11} + 909867 p^{2} T^{12} - 187401 p^{3} T^{13} + 51837 p^{4} T^{14} - 723 p^{6} T^{15} + 2292 p^{6} T^{16} - 336 p^{7} T^{17} + 68 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 12 T + 26 T^{2} - 36 T^{3} + 1143 T^{4} + 5247 T^{5} - 21540 T^{6} - 73476 T^{7} + 337539 T^{8} - 599625 T^{9} - 13374333 T^{10} - 599625 p T^{11} + 337539 p^{2} T^{12} - 73476 p^{3} T^{13} - 21540 p^{4} T^{14} + 5247 p^{5} T^{15} + 1143 p^{6} T^{16} - 36 p^{7} T^{17} + 26 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 3 T + 71 T^{2} + 204 T^{3} + 2646 T^{4} + 8547 T^{5} + 74607 T^{6} + 258954 T^{7} + 1743726 T^{8} + 5989488 T^{9} + 35261703 T^{10} + 5989488 p T^{11} + 1743726 p^{2} T^{12} + 258954 p^{3} T^{13} + 74607 p^{4} T^{14} + 8547 p^{5} T^{15} + 2646 p^{6} T^{16} + 204 p^{7} T^{17} + 71 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 15 T + 195 T^{2} + 1800 T^{3} + 15028 T^{4} + 107121 T^{5} + 712306 T^{6} + 4264599 T^{7} + 24178468 T^{8} + 125954103 T^{9} + 628044835 T^{10} + 125954103 p T^{11} + 24178468 p^{2} T^{12} + 4264599 p^{3} T^{13} + 712306 p^{4} T^{14} + 107121 p^{5} T^{15} + 15028 p^{6} T^{16} + 1800 p^{7} T^{17} + 195 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 15 T + 150 T^{2} + 1125 T^{3} + 6691 T^{4} + 30108 T^{5} + 81631 T^{6} - 232971 T^{7} - 5137202 T^{8} - 44535417 T^{9} - 275752187 T^{10} - 44535417 p T^{11} - 5137202 p^{2} T^{12} - 232971 p^{3} T^{13} + 81631 p^{4} T^{14} + 30108 p^{5} T^{15} + 6691 p^{6} T^{16} + 1125 p^{7} T^{17} + 150 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 7 p T^{2} + 22896 T^{4} - 1551294 T^{6} + 74760504 T^{8} - 2674186965 T^{10} + 74760504 p^{2} T^{12} - 1551294 p^{4} T^{14} + 22896 p^{6} T^{16} - 7 p^{9} T^{18} + p^{10} T^{20} \) | |
| 37 | \( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 10946526 p T^{11} + 10739317 p^{2} T^{12} - 533292 p^{3} T^{13} - 323794 p^{4} T^{14} + 3556 p^{5} T^{15} + 7179 p^{6} T^{16} + 194 p^{7} T^{17} - 97 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 9 T - 34 T^{2} - 747 T^{3} - 2085 T^{4} + 20394 T^{5} + 110775 T^{6} - 15219 p T^{7} - 5992218 T^{8} + 18494757 T^{9} + 381591615 T^{10} + 18494757 p T^{11} - 5992218 p^{2} T^{12} - 15219 p^{4} T^{13} + 110775 p^{4} T^{14} + 20394 p^{5} T^{15} - 2085 p^{6} T^{16} - 747 p^{7} T^{17} - 34 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 3 T - 79 T^{2} + 1100 T^{3} + 1674 T^{4} - 79931 T^{5} + 324899 T^{6} + 75114 p T^{7} - 28512986 T^{8} - 52724394 T^{9} + 1438527201 T^{10} - 52724394 p T^{11} - 28512986 p^{2} T^{12} + 75114 p^{4} T^{13} + 324899 p^{4} T^{14} - 79931 p^{5} T^{15} + 1674 p^{6} T^{16} + 1100 p^{7} T^{17} - 79 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 15 T + 274 T^{2} + 2589 T^{3} + 26755 T^{4} + 176529 T^{5} + 26755 p T^{6} + 2589 p^{2} T^{7} + 274 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 - 9 T + 237 T^{2} - 1890 T^{3} + 28720 T^{4} - 208689 T^{5} + 2523571 T^{6} - 16852668 T^{7} + 178203742 T^{8} - 1088604978 T^{9} + 10361882797 T^{10} - 1088604978 p T^{11} + 178203742 p^{2} T^{12} - 16852668 p^{3} T^{13} + 2523571 p^{4} T^{14} - 208689 p^{5} T^{15} + 28720 p^{6} T^{16} - 1890 p^{7} T^{17} + 237 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( ( 1 - 18 T + 379 T^{2} - 4179 T^{3} + 48448 T^{4} - 365781 T^{5} + 48448 p T^{6} - 4179 p^{2} T^{7} + 379 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 61 | \( 1 - 358 T^{2} + 54069 T^{4} - 4372581 T^{6} + 210774216 T^{8} - 9493721499 T^{10} + 210774216 p^{2} T^{12} - 4372581 p^{4} T^{14} + 54069 p^{6} T^{16} - 358 p^{8} T^{18} + p^{10} T^{20} \) | |
| 67 | \( ( 1 - 10 T + 282 T^{2} - 2634 T^{3} + 34317 T^{4} - 263157 T^{5} + 34317 p T^{6} - 2634 p^{2} T^{7} + 282 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 71 | \( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 747809113 p^{2} T^{12} - 7800935 p^{4} T^{14} + 63037 p^{6} T^{16} - 351 p^{8} T^{18} + p^{10} T^{20} \) | |
| 73 | \( 1 + 3 T + 260 T^{2} + 771 T^{3} + 36639 T^{4} + 155724 T^{5} + 3663555 T^{6} + 21043473 T^{7} + 293486934 T^{8} + 2074196103 T^{9} + 21799556757 T^{10} + 2074196103 p T^{11} + 293486934 p^{2} T^{12} + 21043473 p^{3} T^{13} + 3663555 p^{4} T^{14} + 155724 p^{5} T^{15} + 36639 p^{6} T^{16} + 771 p^{7} T^{17} + 260 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( ( 1 + 20 T + 354 T^{2} + 3468 T^{3} + 34803 T^{4} + 265629 T^{5} + 34803 p T^{6} + 3468 p^{2} T^{7} + 354 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 83 | \( 1 + 15 T - 136 T^{2} - 1773 T^{3} + 22674 T^{4} + 93717 T^{5} - 3687774 T^{6} - 10067337 T^{7} + 346135869 T^{8} + 496605294 T^{9} - 27460905396 T^{10} + 496605294 p T^{11} + 346135869 p^{2} T^{12} - 10067337 p^{3} T^{13} - 3687774 p^{4} T^{14} + 93717 p^{5} T^{15} + 22674 p^{6} T^{16} - 1773 p^{7} T^{17} - 136 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 24 T + 44 T^{2} + 2592 T^{3} - 1287 T^{4} - 278721 T^{5} - 235110 T^{6} + 13705920 T^{7} + 186157425 T^{8} - 904992183 T^{9} - 14602879521 T^{10} - 904992183 p T^{11} + 186157425 p^{2} T^{12} + 13705920 p^{3} T^{13} - 235110 p^{4} T^{14} - 278721 p^{5} T^{15} - 1287 p^{6} T^{16} + 2592 p^{7} T^{17} + 44 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 6 T + 311 T^{2} - 1794 T^{3} + 51903 T^{4} - 385032 T^{5} + 6010353 T^{6} - 63309837 T^{7} + 574248354 T^{8} - 8264282925 T^{9} + 54719955099 T^{10} - 8264282925 p T^{11} + 574248354 p^{2} T^{12} - 63309837 p^{3} T^{13} + 6010353 p^{4} T^{14} - 385032 p^{5} T^{15} + 51903 p^{6} T^{16} - 1794 p^{7} T^{17} + 311 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.01857812999223005696508685715, −3.97072784669111082360687724284, −3.88903305288852265663357576810, −3.80569675637018367650430107868, −3.64501999721411538706970235182, −3.63017630275556704851826796779, −3.43932382490988152209665734474, −3.35937815628962683166431625128, −3.06093404620130424883308998611, −2.95178901499660369923101562435, −2.81190724285823450683648968297, −2.69268092309375445616880933042, −2.53143192101435435987631866293, −2.42866458747154505695985043903, −2.36398795213447887577533666323, −2.31400149155128380771377675041, −2.28371162782614783119735658250, −2.18099534891046402330818123372, −1.70764027146353197247708481128, −1.70298781048515246146949504575, −1.55035418268486757192659673714, −1.35609211620995449642188996282, −1.18102615293436640557812269561, −0.59550883979910780313247642802, −0.20763361549790341593987244868, 0.20763361549790341593987244868, 0.59550883979910780313247642802, 1.18102615293436640557812269561, 1.35609211620995449642188996282, 1.55035418268486757192659673714, 1.70298781048515246146949504575, 1.70764027146353197247708481128, 2.18099534891046402330818123372, 2.28371162782614783119735658250, 2.31400149155128380771377675041, 2.36398795213447887577533666323, 2.42866458747154505695985043903, 2.53143192101435435987631866293, 2.69268092309375445616880933042, 2.81190724285823450683648968297, 2.95178901499660369923101562435, 3.06093404620130424883308998611, 3.35937815628962683166431625128, 3.43932382490988152209665734474, 3.63017630275556704851826796779, 3.64501999721411538706970235182, 3.80569675637018367650430107868, 3.88903305288852265663357576810, 3.97072784669111082360687724284, 4.01857812999223005696508685715
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.