Dirichlet series
| L(s) = 1 | + 2·2-s − 3·4-s − 10·5-s − 8·8-s − 10·9-s − 20·10-s − 10·11-s − 8·13-s + 16-s − 28·17-s − 20·18-s − 8·19-s + 30·20-s − 20·22-s − 8·23-s + 55·25-s − 16·26-s − 8·29-s + 4·31-s + 10·32-s − 56·34-s + 30·36-s + 28·37-s − 16·38-s + 80·40-s − 44·41-s + 20·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 3/2·4-s − 4.47·5-s − 2.82·8-s − 3.33·9-s − 6.32·10-s − 3.01·11-s − 2.21·13-s + 1/4·16-s − 6.79·17-s − 4.71·18-s − 1.83·19-s + 6.70·20-s − 4.26·22-s − 1.66·23-s + 11·25-s − 3.13·26-s − 1.48·29-s + 0.718·31-s + 1.76·32-s − 9.60·34-s + 5·36-s + 4.60·37-s − 2.59·38-s + 12.6·40-s − 6.87·41-s + 3.04·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(5^{10} \cdot 7^{20} \cdot 11^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.12989\times 10^{13}\) |
| Root analytic conductor: | \(4.63893\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 5^{10} \cdot 7^{20} \cdot 11^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T )^{10} \) |
| 7 | \( 1 \) | |
| 11 | \( ( 1 + T )^{10} \) | |
| good | 2 | \( 1 - p T + 7 T^{2} - 3 p^{2} T^{3} + 7 p^{2} T^{4} - 11 p^{2} T^{5} + 45 p T^{6} - 33 p^{2} T^{7} + 239 T^{8} - 159 p T^{9} + 525 T^{10} - 159 p^{2} T^{11} + 239 p^{2} T^{12} - 33 p^{5} T^{13} + 45 p^{5} T^{14} - 11 p^{7} T^{15} + 7 p^{8} T^{16} - 3 p^{9} T^{17} + 7 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 + 10 T^{2} + 55 T^{4} - 4 p T^{5} + 232 T^{6} - 112 T^{7} + 806 T^{8} - 472 T^{9} + 2482 T^{10} - 472 p T^{11} + 806 p^{2} T^{12} - 112 p^{3} T^{13} + 232 p^{4} T^{14} - 4 p^{6} T^{15} + 55 p^{6} T^{16} + 10 p^{8} T^{18} + p^{10} T^{20} \) | |
| 13 | \( 1 + 8 T + 88 T^{2} + 488 T^{3} + 3155 T^{4} + 13428 T^{5} + 66702 T^{6} + 236508 T^{7} + 1033330 T^{8} + 3322916 T^{9} + 13900258 T^{10} + 3322916 p T^{11} + 1033330 p^{2} T^{12} + 236508 p^{3} T^{13} + 66702 p^{4} T^{14} + 13428 p^{5} T^{15} + 3155 p^{6} T^{16} + 488 p^{7} T^{17} + 88 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 28 T + 468 T^{2} + 5696 T^{3} + 55563 T^{4} + 453676 T^{5} + 3190340 T^{6} + 19645224 T^{7} + 107154258 T^{8} + 521194244 T^{9} + 133520286 p T^{10} + 521194244 p T^{11} + 107154258 p^{2} T^{12} + 19645224 p^{3} T^{13} + 3190340 p^{4} T^{14} + 453676 p^{5} T^{15} + 55563 p^{6} T^{16} + 5696 p^{7} T^{17} + 468 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 8 T + 148 T^{2} + 976 T^{3} + 10069 T^{4} + 57376 T^{5} + 427912 T^{6} + 2141312 T^{7} + 12746178 T^{8} + 55958696 T^{9} + 279920560 T^{10} + 55958696 p T^{11} + 12746178 p^{2} T^{12} + 2141312 p^{3} T^{13} + 427912 p^{4} T^{14} + 57376 p^{5} T^{15} + 10069 p^{6} T^{16} + 976 p^{7} T^{17} + 148 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 8 T + 160 T^{2} + 1016 T^{3} + 12059 T^{4} + 65588 T^{5} + 589040 T^{6} + 2807900 T^{7} + 20687624 T^{8} + 86594832 T^{9} + 23706596 p T^{10} + 86594832 p T^{11} + 20687624 p^{2} T^{12} + 2807900 p^{3} T^{13} + 589040 p^{4} T^{14} + 65588 p^{5} T^{15} + 12059 p^{6} T^{16} + 1016 p^{7} T^{17} + 160 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 8 T + 190 T^{2} + 1432 T^{3} + 18001 T^{4} + 123272 T^{5} + 1113904 T^{6} + 6770360 T^{7} + 49597270 T^{8} + 264366240 T^{9} + 1653345780 T^{10} + 264366240 p T^{11} + 49597270 p^{2} T^{12} + 6770360 p^{3} T^{13} + 1113904 p^{4} T^{14} + 123272 p^{5} T^{15} + 18001 p^{6} T^{16} + 1432 p^{7} T^{17} + 190 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 4 T + 180 T^{2} - 996 T^{3} + 16985 T^{4} - 101972 T^{5} + 1105426 T^{6} - 6341132 T^{7} + 52402230 T^{8} - 274330360 T^{9} + 1861207142 T^{10} - 274330360 p T^{11} + 52402230 p^{2} T^{12} - 6341132 p^{3} T^{13} + 1105426 p^{4} T^{14} - 101972 p^{5} T^{15} + 16985 p^{6} T^{16} - 996 p^{7} T^{17} + 180 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 28 T + 650 T^{2} - 10316 T^{3} + 142911 T^{4} - 1618748 T^{5} + 16441044 T^{6} - 144374260 T^{7} + 1152513136 T^{8} - 8130559800 T^{9} + 52450348376 T^{10} - 8130559800 p T^{11} + 1152513136 p^{2} T^{12} - 144374260 p^{3} T^{13} + 16441044 p^{4} T^{14} - 1618748 p^{5} T^{15} + 142911 p^{6} T^{16} - 10316 p^{7} T^{17} + 650 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 44 T + 1114 T^{2} + 20492 T^{3} + 302341 T^{4} + 3745684 T^{5} + 40104878 T^{6} + 377562004 T^{7} + 3162268482 T^{8} + 23723587864 T^{9} + 160082916450 T^{10} + 23723587864 p T^{11} + 3162268482 p^{2} T^{12} + 377562004 p^{3} T^{13} + 40104878 p^{4} T^{14} + 3745684 p^{5} T^{15} + 302341 p^{6} T^{16} + 20492 p^{7} T^{17} + 1114 p^{8} T^{18} + 44 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 20 T + 394 T^{2} - 5164 T^{3} + 63747 T^{4} - 649068 T^{5} + 6229020 T^{6} - 52849604 T^{7} + 423767028 T^{8} - 3079393872 T^{9} + 21145722872 T^{10} - 3079393872 p T^{11} + 423767028 p^{2} T^{12} - 52849604 p^{3} T^{13} + 6229020 p^{4} T^{14} - 649068 p^{5} T^{15} + 63747 p^{6} T^{16} - 5164 p^{7} T^{17} + 394 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 12 T + 336 T^{2} + 3368 T^{3} + 53551 T^{4} + 467328 T^{5} + 5495722 T^{6} + 42023728 T^{7} + 402083906 T^{8} + 2683013616 T^{9} + 21815206330 T^{10} + 2683013616 p T^{11} + 402083906 p^{2} T^{12} + 42023728 p^{3} T^{13} + 5495722 p^{4} T^{14} + 467328 p^{5} T^{15} + 53551 p^{6} T^{16} + 3368 p^{7} T^{17} + 336 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 170 T^{2} + 500 T^{3} + 16195 T^{4} + 74400 T^{5} + 1317672 T^{6} + 5715080 T^{7} + 90306012 T^{8} + 358734388 T^{9} + 5144619536 T^{10} + 358734388 p T^{11} + 90306012 p^{2} T^{12} + 5715080 p^{3} T^{13} + 1317672 p^{4} T^{14} + 74400 p^{5} T^{15} + 16195 p^{6} T^{16} + 500 p^{7} T^{17} + 170 p^{8} T^{18} + p^{10} T^{20} \) | |
| 59 | \( 1 + 16 T + 462 T^{2} + 5456 T^{3} + 89669 T^{4} + 830228 T^{5} + 10178630 T^{6} + 77727980 T^{7} + 807529846 T^{8} + 5406386904 T^{9} + 51490587202 T^{10} + 5406386904 p T^{11} + 807529846 p^{2} T^{12} + 77727980 p^{3} T^{13} + 10178630 p^{4} T^{14} + 830228 p^{5} T^{15} + 89669 p^{6} T^{16} + 5456 p^{7} T^{17} + 462 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 16 T + 430 T^{2} + 4340 T^{3} + 70841 T^{4} + 498608 T^{5} + 6595716 T^{6} + 32062512 T^{7} + 426517574 T^{8} + 1491048756 T^{9} + 25018038894 T^{10} + 1491048756 p T^{11} + 426517574 p^{2} T^{12} + 32062512 p^{3} T^{13} + 6595716 p^{4} T^{14} + 498608 p^{5} T^{15} + 70841 p^{6} T^{16} + 4340 p^{7} T^{17} + 430 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 20 T + 714 T^{2} - 10792 T^{3} + 218555 T^{4} - 2648064 T^{5} + 38915680 T^{6} - 389468984 T^{7} + 4531361972 T^{8} - 37940269580 T^{9} + 363208007712 T^{10} - 37940269580 p T^{11} + 4531361972 p^{2} T^{12} - 389468984 p^{3} T^{13} + 38915680 p^{4} T^{14} - 2648064 p^{5} T^{15} + 218555 p^{6} T^{16} - 10792 p^{7} T^{17} + 714 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 4 T + 586 T^{2} + 2044 T^{3} + 160997 T^{4} + 488488 T^{5} + 27335480 T^{6} + 71928760 T^{7} + 3179772234 T^{8} + 7214939880 T^{9} + 265002161164 T^{10} + 7214939880 p T^{11} + 3179772234 p^{2} T^{12} + 71928760 p^{3} T^{13} + 27335480 p^{4} T^{14} + 488488 p^{5} T^{15} + 160997 p^{6} T^{16} + 2044 p^{7} T^{17} + 586 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 20 T + 530 T^{2} + 8124 T^{3} + 135367 T^{4} + 1682036 T^{5} + 21774848 T^{6} + 230065792 T^{7} + 2476162390 T^{8} + 22597901824 T^{9} + 208778394474 T^{10} + 22597901824 p T^{11} + 2476162390 p^{2} T^{12} + 230065792 p^{3} T^{13} + 21774848 p^{4} T^{14} + 1682036 p^{5} T^{15} + 135367 p^{6} T^{16} + 8124 p^{7} T^{17} + 530 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 20 T + 558 T^{2} + 8980 T^{3} + 152845 T^{4} + 1992524 T^{5} + 26239744 T^{6} + 288900828 T^{7} + 3173198134 T^{8} + 30252363160 T^{9} + 287236111424 T^{10} + 30252363160 p T^{11} + 3173198134 p^{2} T^{12} + 288900828 p^{3} T^{13} + 26239744 p^{4} T^{14} + 1992524 p^{5} T^{15} + 152845 p^{6} T^{16} + 8980 p^{7} T^{17} + 558 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 16 T + 444 T^{2} + 5492 T^{3} + 100813 T^{4} + 1140580 T^{5} + 16001868 T^{6} + 161232372 T^{7} + 1878038098 T^{8} + 17285606436 T^{9} + 175836262408 T^{10} + 17285606436 p T^{11} + 1878038098 p^{2} T^{12} + 161232372 p^{3} T^{13} + 16001868 p^{4} T^{14} + 1140580 p^{5} T^{15} + 100813 p^{6} T^{16} + 5492 p^{7} T^{17} + 444 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 44 T + 1356 T^{2} + 31476 T^{3} + 612413 T^{4} + 10234528 T^{5} + 151181176 T^{6} + 1996280608 T^{7} + 23820011458 T^{8} + 258145925600 T^{9} + 2549769918240 T^{10} + 258145925600 p T^{11} + 23820011458 p^{2} T^{12} + 1996280608 p^{3} T^{13} + 151181176 p^{4} T^{14} + 10234528 p^{5} T^{15} + 612413 p^{6} T^{16} + 31476 p^{7} T^{17} + 1356 p^{8} T^{18} + 44 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 4 T + 490 T^{2} - 1440 T^{3} + 102505 T^{4} - 225860 T^{5} + 13218484 T^{6} - 28047068 T^{7} + 1387661302 T^{8} - 3546336220 T^{9} + 1422507532 p T^{10} - 3546336220 p T^{11} + 1387661302 p^{2} T^{12} - 28047068 p^{3} T^{13} + 13218484 p^{4} T^{14} - 225860 p^{5} T^{15} + 102505 p^{6} T^{16} - 1440 p^{7} T^{17} + 490 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.70050832443028829569983403620, −3.62995411565619449083478877880, −3.39849113588686722432291315713, −3.25214296739737709340751748369, −3.11760342843965074779477869619, −3.08723939327633545479733782422, −2.99603826907860840292470022476, −2.95031615013106161229652085885, −2.79789994334040173678068503114, −2.72859769185163674276037329912, −2.65030759797813225051809826506, −2.64289625628216143060505292706, −2.54179993955079052857380990725, −2.42352871133694003098679198465, −2.33126691699956171924129784004, −2.14866648595135544503789652091, −2.08329649789993858756052081627, −1.90468071928692373376052901981, −1.90407072163281678524231327701, −1.82758893974264473072775452628, −1.48398293947217292054912099066, −1.18578400504096405892389115847, −1.10873826481534212582072793618, −1.02094199391417260209716938105, −0.913137272040709892547366962596, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.913137272040709892547366962596, 1.02094199391417260209716938105, 1.10873826481534212582072793618, 1.18578400504096405892389115847, 1.48398293947217292054912099066, 1.82758893974264473072775452628, 1.90407072163281678524231327701, 1.90468071928692373376052901981, 2.08329649789993858756052081627, 2.14866648595135544503789652091, 2.33126691699956171924129784004, 2.42352871133694003098679198465, 2.54179993955079052857380990725, 2.64289625628216143060505292706, 2.65030759797813225051809826506, 2.72859769185163674276037329912, 2.79789994334040173678068503114, 2.95031615013106161229652085885, 2.99603826907860840292470022476, 3.08723939327633545479733782422, 3.11760342843965074779477869619, 3.25214296739737709340751748369, 3.39849113588686722432291315713, 3.62995411565619449083478877880, 3.70050832443028829569983403620
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.