Dirichlet series
| L(s) = 1 | − 4·2-s + 3-s + 2·4-s − 4·5-s − 4·6-s + 14·8-s + 6·9-s + 16·10-s + 4·11-s + 2·12-s + 8·13-s − 4·15-s − 27·16-s − 12·17-s − 24·18-s − 19-s − 8·20-s − 16·22-s + 3·23-s + 14·24-s + 20·25-s − 32·26-s + 8·27-s + 7·29-s + 16·30-s − 6·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.577·3-s + 4-s − 1.78·5-s − 1.63·6-s + 4.94·8-s + 2·9-s + 5.05·10-s + 1.20·11-s + 0.577·12-s + 2.21·13-s − 1.03·15-s − 6.75·16-s − 2.91·17-s − 5.65·18-s − 0.229·19-s − 1.78·20-s − 3.41·22-s + 0.625·23-s + 2.85·24-s + 4·25-s − 6.27·26-s + 1.53·27-s + 1.29·29-s + 2.92·30-s − 1.07·31-s + 1.41·32-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
| \(L(1)\) | \(\approx\) | \(0.2736497418\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2736497418\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - T - 5 T^{2} + p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} + p^{3} T^{7} - 5 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) |
| 7 | \( 1 \) | |
| good | 2 | \( ( 1 + p T + 5 T^{2} + 7 T^{3} + 13 T^{4} + 15 T^{5} + 13 p T^{6} + 7 p^{2} T^{7} + 5 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} )^{2} \) |
| 5 | \( 1 + 4 T - 4 T^{2} - 44 T^{3} - 41 T^{4} + 119 T^{5} + 222 T^{6} + 456 T^{7} + 1623 T^{8} - 2021 T^{9} - 16541 T^{10} - 2021 p T^{11} + 1623 p^{2} T^{12} + 456 p^{3} T^{13} + 222 p^{4} T^{14} + 119 p^{5} T^{15} - 41 p^{6} T^{16} - 44 p^{7} T^{17} - 4 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 135733 p T^{11} + 98940 p^{2} T^{12} + 27495 p^{3} T^{13} - 8385 p^{4} T^{14} - 2492 p^{5} T^{15} + 607 p^{6} T^{16} + 134 p^{7} T^{17} - 31 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 8 T - 14 T^{2} + 14 p T^{3} + 686 T^{4} - 4429 T^{5} - 12871 T^{6} + 3323 p T^{7} + 305249 T^{8} - 358672 T^{9} - 3841969 T^{10} - 358672 p T^{11} + 305249 p^{2} T^{12} + 3323 p^{4} T^{13} - 12871 p^{4} T^{14} - 4429 p^{5} T^{15} + 686 p^{6} T^{16} + 14 p^{8} T^{17} - 14 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 12 T + 14 T^{2} - 192 T^{3} + 1185 T^{4} + 11847 T^{5} - 6180 T^{6} - 65736 T^{7} + 1002861 T^{8} + 2436261 T^{9} - 7749777 T^{10} + 2436261 p T^{11} + 1002861 p^{2} T^{12} - 65736 p^{3} T^{13} - 6180 p^{4} T^{14} + 11847 p^{5} T^{15} + 1185 p^{6} T^{16} - 192 p^{7} T^{17} + 14 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + T - 53 T^{2} - 10 p T^{3} + 1262 T^{4} + 7007 T^{5} - 13111 T^{6} - 116110 T^{7} + 67964 T^{8} + 721616 T^{9} - 440023 T^{10} + 721616 p T^{11} + 67964 p^{2} T^{12} - 116110 p^{3} T^{13} - 13111 p^{4} T^{14} + 7007 p^{5} T^{15} + 1262 p^{6} T^{16} - 10 p^{8} T^{17} - 53 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 3271749 p T^{11} + 502362 p^{2} T^{12} - 135027 p^{3} T^{13} + 21792 p^{4} T^{14} - 5127 p^{5} T^{15} + 6 p^{6} T^{16} + 294 p^{7} T^{17} - 43 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 2570041 p T^{11} + 6918636 p^{2} T^{12} + 341373 p^{3} T^{13} - 199563 p^{4} T^{14} - 15146 p^{5} T^{15} + 4561 p^{6} T^{16} + 419 p^{7} T^{17} - 76 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( ( 1 + 3 T + 134 T^{2} + 308 T^{3} + 250 p T^{4} + 13615 T^{5} + 250 p^{2} T^{6} + 308 p^{2} T^{7} + 134 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 37 | \( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 33131096 p T^{11} - 9801827 p^{2} T^{12} + 2296536 p^{3} T^{13} + 27130 p^{4} T^{14} - 45352 p^{5} T^{15} + 4503 p^{6} T^{16} + 560 p^{7} T^{17} - 89 p^{8} T^{18} + p^{10} T^{20} \) | |
| 41 | \( 1 + 5 T - 136 T^{2} - 733 T^{3} + 10507 T^{4} + 54412 T^{5} - 554055 T^{6} - 2345451 T^{7} + 23706084 T^{8} + 41392439 T^{9} - 952045937 T^{10} + 41392439 p T^{11} + 23706084 p^{2} T^{12} - 2345451 p^{3} T^{13} - 554055 p^{4} T^{14} + 54412 p^{5} T^{15} + 10507 p^{6} T^{16} - 733 p^{7} T^{17} - 136 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 15416324 p T^{11} - 3708600 p^{2} T^{12} + 1632678 p^{3} T^{13} - 95427 p^{4} T^{14} - 3843 p^{5} T^{15} + 7014 p^{6} T^{16} - 66 p^{7} T^{17} - 77 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 - 27 T + 448 T^{2} - 5169 T^{3} + 48091 T^{4} - 359985 T^{5} + 48091 p T^{6} - 5169 p^{2} T^{7} + 448 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 340111866 p T^{11} + 110973258 p^{2} T^{12} - 3783942 p^{3} T^{13} - 601755 p^{4} T^{14} + 177027 p^{5} T^{15} + 12966 p^{6} T^{16} - 924 p^{7} T^{17} + 41 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( ( 1 - 30 T + 601 T^{2} - 8193 T^{3} + 88864 T^{4} - 752289 T^{5} + 88864 p T^{6} - 8193 p^{2} T^{7} + 601 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 61 | \( ( 1 + 14 T + 339 T^{2} + 3409 T^{3} + 43418 T^{4} + 311709 T^{5} + 43418 p T^{6} + 3409 p^{2} T^{7} + 339 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 67 | \( ( 1 - 2 T + 132 T^{2} - 196 T^{3} + 10871 T^{4} - 15429 T^{5} + 10871 p T^{6} - 196 p^{2} T^{7} + 132 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 71 | \( ( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 15679 p T^{6} + 285 p^{2} T^{7} + 187 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 + 15 T - 134 T^{2} - 2501 T^{3} + 16563 T^{4} + 235276 T^{5} - 2002535 T^{6} - 9021201 T^{7} + 288508378 T^{8} + 238799411 T^{9} - 25271949561 T^{10} + 238799411 p T^{11} + 288508378 p^{2} T^{12} - 9021201 p^{3} T^{13} - 2002535 p^{4} T^{14} + 235276 p^{5} T^{15} + 16563 p^{6} T^{16} - 2501 p^{7} T^{17} - 134 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( ( 1 - 4 T + 300 T^{2} - 1488 T^{3} + 39873 T^{4} - 184983 T^{5} + 39873 p T^{6} - 1488 p^{2} T^{7} + 300 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 83 | \( 1 + 9 T - 148 T^{2} + 297 T^{3} + 24654 T^{4} - 118125 T^{5} - 807174 T^{6} + 21382137 T^{7} - 37648479 T^{8} - 452536146 T^{9} + 15509586612 T^{10} - 452536146 p T^{11} - 37648479 p^{2} T^{12} + 21382137 p^{3} T^{13} - 807174 p^{4} T^{14} - 118125 p^{5} T^{15} + 24654 p^{6} T^{16} + 297 p^{7} T^{17} - 148 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 28 T + 104 T^{2} - 1736 T^{3} + 31273 T^{4} + 611939 T^{5} - 1780638 T^{6} - 18973932 T^{7} + 740914101 T^{8} + 3271180573 T^{9} - 40614588329 T^{10} + 3271180573 p T^{11} + 740914101 p^{2} T^{12} - 18973932 p^{3} T^{13} - 1780638 p^{4} T^{14} + 611939 p^{5} T^{15} + 31273 p^{6} T^{16} - 1736 p^{7} T^{17} + 104 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 12 T - 197 T^{2} + 1534 T^{3} + 27813 T^{4} - 14090 T^{5} - 4545035 T^{6} + 6881349 T^{7} + 472663750 T^{8} - 908843245 T^{9} - 38512186359 T^{10} - 908843245 p T^{11} + 472663750 p^{2} T^{12} + 6881349 p^{3} T^{13} - 4545035 p^{4} T^{14} - 14090 p^{5} T^{15} + 27813 p^{6} T^{16} + 1534 p^{7} T^{17} - 197 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.22384326682770742445562217467, −4.20752881099597758829632750848, −3.98491163037614810955031097286, −3.86210688314794208280573218652, −3.84534710279292482956267870239, −3.82974495078955425086750790604, −3.66284566054068512990272851749, −3.25453694760961284341379274111, −3.17312869262596026090421207065, −2.98271630836785874010859626770, −2.97486068629253709505070729674, −2.83794593389116415653035430926, −2.81133418574209614392623897199, −2.30207758835454709085545040551, −2.26319804427224440741408921516, −2.12273432553517660731884850959, −2.01816875309205544487189948943, −1.59250992717026269522047426366, −1.53943768469215698140188523372, −1.45163335863793393107762228213, −0.989287337910654548576680380373, −0.860738025715233225981353269194, −0.74649135379333170155553149136, −0.73532301455088347258011576102, −0.25186025295442320195727123095, 0.25186025295442320195727123095, 0.73532301455088347258011576102, 0.74649135379333170155553149136, 0.860738025715233225981353269194, 0.989287337910654548576680380373, 1.45163335863793393107762228213, 1.53943768469215698140188523372, 1.59250992717026269522047426366, 2.01816875309205544487189948943, 2.12273432553517660731884850959, 2.26319804427224440741408921516, 2.30207758835454709085545040551, 2.81133418574209614392623897199, 2.83794593389116415653035430926, 2.97486068629253709505070729674, 2.98271630836785874010859626770, 3.17312869262596026090421207065, 3.25453694760961284341379274111, 3.66284566054068512990272851749, 3.82974495078955425086750790604, 3.84534710279292482956267870239, 3.86210688314794208280573218652, 3.98491163037614810955031097286, 4.20752881099597758829632750848, 4.22384326682770742445562217467
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.