Dirichlet series
| L(s) = 1 | − 2·3-s − 4·5-s − 8·9-s − 6·13-s + 8·15-s − 12·17-s − 8·19-s + 4·23-s − 9·25-s + 18·27-s + 2·29-s − 8·31-s − 2·37-s + 12·39-s − 10·41-s − 2·43-s + 32·45-s + 14·47-s + 24·51-s + 8·53-s + 16·57-s − 24·59-s − 14·61-s + 24·65-s − 8·67-s − 8·69-s + 10·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s − 8/3·9-s − 1.66·13-s + 2.06·15-s − 2.91·17-s − 1.83·19-s + 0.834·23-s − 9/5·25-s + 3.46·27-s + 0.371·29-s − 1.43·31-s − 0.328·37-s + 1.92·39-s − 1.56·41-s − 0.304·43-s + 4.77·45-s + 2.04·47-s + 3.36·51-s + 1.09·53-s + 2.11·57-s − 3.12·59-s − 1.79·61-s + 2.97·65-s − 0.977·67-s − 0.963·69-s + 1.18·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 7^{20} \cdot 41^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18350\times 10^{18}\) |
| Root analytic conductor: | \(8.01047\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 2^{20} \cdot 7^{20} \cdot 41^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
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 | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| 41 | \( ( 1 + T )^{10} \) | |
| good | 3 | \( 1 + 2 T + 4 p T^{2} + 22 T^{3} + 83 T^{4} + 130 T^{5} + 134 p T^{6} + 542 T^{7} + 1513 T^{8} + 206 p^{2} T^{9} + 4817 T^{10} + 206 p^{3} T^{11} + 1513 p^{2} T^{12} + 542 p^{3} T^{13} + 134 p^{5} T^{14} + 130 p^{5} T^{15} + 83 p^{6} T^{16} + 22 p^{7} T^{17} + 4 p^{9} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + 4 T + p^{2} T^{2} + 88 T^{3} + 326 T^{4} + 994 T^{5} + 2938 T^{6} + 7978 T^{7} + 4149 p T^{8} + 50112 T^{9} + 116858 T^{10} + 50112 p T^{11} + 4149 p^{3} T^{12} + 7978 p^{3} T^{13} + 2938 p^{4} T^{14} + 994 p^{5} T^{15} + 326 p^{6} T^{16} + 88 p^{7} T^{17} + p^{10} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 60 T^{2} - 48 T^{3} + 1750 T^{4} - 2512 T^{5} + 3186 p T^{6} - 60128 T^{7} + 548345 T^{8} - 910432 T^{9} + 6806204 T^{10} - 910432 p T^{11} + 548345 p^{2} T^{12} - 60128 p^{3} T^{13} + 3186 p^{5} T^{14} - 2512 p^{5} T^{15} + 1750 p^{6} T^{16} - 48 p^{7} T^{17} + 60 p^{8} T^{18} + p^{10} T^{20} \) | |
| 13 | \( 1 + 6 T + 77 T^{2} + 378 T^{3} + 212 p T^{4} + 11378 T^{5} + 63254 T^{6} + 1338 p^{2} T^{7} + 1083085 T^{8} + 3478468 T^{9} + 15235973 T^{10} + 3478468 p T^{11} + 1083085 p^{2} T^{12} + 1338 p^{5} T^{13} + 63254 p^{4} T^{14} + 11378 p^{5} T^{15} + 212 p^{7} T^{16} + 378 p^{7} T^{17} + 77 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 12 T + 9 p T^{2} + 1250 T^{3} + 9866 T^{4} + 64080 T^{5} + 390146 T^{6} + 2122674 T^{7} + 10664977 T^{8} + 49516210 T^{9} + 211311741 T^{10} + 49516210 p T^{11} + 10664977 p^{2} T^{12} + 2122674 p^{3} T^{13} + 390146 p^{4} T^{14} + 64080 p^{5} T^{15} + 9866 p^{6} T^{16} + 1250 p^{7} T^{17} + 9 p^{9} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 8 T + 142 T^{2} + 854 T^{3} + 8779 T^{4} + 41722 T^{5} + 324726 T^{6} + 1272034 T^{7} + 8468743 T^{8} + 28827300 T^{9} + 175736485 T^{10} + 28827300 p T^{11} + 8468743 p^{2} T^{12} + 1272034 p^{3} T^{13} + 324726 p^{4} T^{14} + 41722 p^{5} T^{15} + 8779 p^{6} T^{16} + 854 p^{7} T^{17} + 142 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 4 T + 150 T^{2} - 472 T^{3} + 10995 T^{4} - 29014 T^{5} + 530388 T^{6} - 1210146 T^{7} + 18532865 T^{8} - 36844898 T^{9} + 487837739 T^{10} - 36844898 p T^{11} + 18532865 p^{2} T^{12} - 1210146 p^{3} T^{13} + 530388 p^{4} T^{14} - 29014 p^{5} T^{15} + 10995 p^{6} T^{16} - 472 p^{7} T^{17} + 150 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 2 T + 169 T^{2} - 200 T^{3} + 13298 T^{4} - 8180 T^{5} + 689398 T^{6} - 273872 T^{7} + 27573885 T^{8} - 10844658 T^{9} + 890329250 T^{10} - 10844658 p T^{11} + 27573885 p^{2} T^{12} - 273872 p^{3} T^{13} + 689398 p^{4} T^{14} - 8180 p^{5} T^{15} + 13298 p^{6} T^{16} - 200 p^{7} T^{17} + 169 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 8 T + 173 T^{2} + 1172 T^{3} + 15518 T^{4} + 93450 T^{5} + 940146 T^{6} + 5050870 T^{7} + 42210833 T^{8} + 203386776 T^{9} + 1475895842 T^{10} + 203386776 p T^{11} + 42210833 p^{2} T^{12} + 5050870 p^{3} T^{13} + 940146 p^{4} T^{14} + 93450 p^{5} T^{15} + 15518 p^{6} T^{16} + 1172 p^{7} T^{17} + 173 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 2 T + 207 T^{2} + 440 T^{3} + 23279 T^{4} + 49382 T^{5} + 1758529 T^{6} + 3562350 T^{7} + 97635520 T^{8} + 181110266 T^{9} + 4119614704 T^{10} + 181110266 p T^{11} + 97635520 p^{2} T^{12} + 3562350 p^{3} T^{13} + 1758529 p^{4} T^{14} + 49382 p^{5} T^{15} + 23279 p^{6} T^{16} + 440 p^{7} T^{17} + 207 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 2 T + 221 T^{2} + 468 T^{3} + 25826 T^{4} + 55060 T^{5} + 2056468 T^{6} + 4339622 T^{7} + 123599993 T^{8} + 247838392 T^{9} + 5913403015 T^{10} + 247838392 p T^{11} + 123599993 p^{2} T^{12} + 4339622 p^{3} T^{13} + 2056468 p^{4} T^{14} + 55060 p^{5} T^{15} + 25826 p^{6} T^{16} + 468 p^{7} T^{17} + 221 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 14 T + 347 T^{2} - 4286 T^{3} + 61213 T^{4} - 13504 p T^{5} + 6842191 T^{6} - 59787324 T^{7} + 525899550 T^{8} - 3920374812 T^{9} + 28988297588 T^{10} - 3920374812 p T^{11} + 525899550 p^{2} T^{12} - 59787324 p^{3} T^{13} + 6842191 p^{4} T^{14} - 13504 p^{6} T^{15} + 61213 p^{6} T^{16} - 4286 p^{7} T^{17} + 347 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 8 T + 245 T^{2} - 1196 T^{3} + 23954 T^{4} - 58632 T^{5} + 1591642 T^{6} - 2732816 T^{7} + 116936509 T^{8} - 293998276 T^{9} + 7553425474 T^{10} - 293998276 p T^{11} + 116936509 p^{2} T^{12} - 2732816 p^{3} T^{13} + 1591642 p^{4} T^{14} - 58632 p^{5} T^{15} + 23954 p^{6} T^{16} - 1196 p^{7} T^{17} + 245 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 24 T + 629 T^{2} + 8776 T^{3} + 126918 T^{4} + 1161538 T^{5} + 11387310 T^{6} + 65476022 T^{7} + 482347017 T^{8} + 1374681036 T^{9} + 15949341018 T^{10} + 1374681036 p T^{11} + 482347017 p^{2} T^{12} + 65476022 p^{3} T^{13} + 11387310 p^{4} T^{14} + 1161538 p^{5} T^{15} + 126918 p^{6} T^{16} + 8776 p^{7} T^{17} + 629 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 14 T + 528 T^{2} + 6362 T^{3} + 130430 T^{4} + 1349402 T^{5} + 19744598 T^{6} + 175069242 T^{7} + 2021257345 T^{8} + 15296907412 T^{9} + 145932869524 T^{10} + 15296907412 p T^{11} + 2021257345 p^{2} T^{12} + 175069242 p^{3} T^{13} + 19744598 p^{4} T^{14} + 1349402 p^{5} T^{15} + 130430 p^{6} T^{16} + 6362 p^{7} T^{17} + 528 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 8 T + 449 T^{2} + 2320 T^{3} + 87078 T^{4} + 265904 T^{5} + 10436462 T^{6} + 16581400 T^{7} + 935678857 T^{8} + 796918512 T^{9} + 68666864514 T^{10} + 796918512 p T^{11} + 935678857 p^{2} T^{12} + 16581400 p^{3} T^{13} + 10436462 p^{4} T^{14} + 265904 p^{5} T^{15} + 87078 p^{6} T^{16} + 2320 p^{7} T^{17} + 449 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 10 T + 444 T^{2} - 4262 T^{3} + 86730 T^{4} - 810086 T^{5} + 9918442 T^{6} - 94008490 T^{7} + 789911765 T^{8} - 7996076064 T^{9} + 55295925748 T^{10} - 7996076064 p T^{11} + 789911765 p^{2} T^{12} - 94008490 p^{3} T^{13} + 9918442 p^{4} T^{14} - 810086 p^{5} T^{15} + 86730 p^{6} T^{16} - 4262 p^{7} T^{17} + 444 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 44 T + 1220 T^{2} + 24508 T^{3} + 403542 T^{4} + 5667300 T^{5} + 70965058 T^{6} + 804708756 T^{7} + 8403572217 T^{8} + 80839769728 T^{9} + 719421847316 T^{10} + 80839769728 p T^{11} + 8403572217 p^{2} T^{12} + 804708756 p^{3} T^{13} + 70965058 p^{4} T^{14} + 5667300 p^{5} T^{15} + 403542 p^{6} T^{16} + 24508 p^{7} T^{17} + 1220 p^{8} T^{18} + 44 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 10 T + 614 T^{2} - 5902 T^{3} + 179069 T^{4} - 1630352 T^{5} + 32727400 T^{6} - 275265280 T^{7} + 4153988066 T^{8} - 31232651988 T^{9} + 382732416868 T^{10} - 31232651988 p T^{11} + 4153988066 p^{2} T^{12} - 275265280 p^{3} T^{13} + 32727400 p^{4} T^{14} - 1630352 p^{5} T^{15} + 179069 p^{6} T^{16} - 5902 p^{7} T^{17} + 614 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 20 T + 528 T^{2} + 7496 T^{3} + 121918 T^{4} + 1442276 T^{5} + 18880926 T^{6} + 198399444 T^{7} + 2251070337 T^{8} + 21080323924 T^{9} + 210699872260 T^{10} + 21080323924 p T^{11} + 2251070337 p^{2} T^{12} + 198399444 p^{3} T^{13} + 18880926 p^{4} T^{14} + 1442276 p^{5} T^{15} + 121918 p^{6} T^{16} + 7496 p^{7} T^{17} + 528 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 6 T + 2 p T^{2} + 40 T^{3} + 19025 T^{4} - 13312 T^{5} + 2644880 T^{6} + 1951728 T^{7} + 298984707 T^{8} - 249612878 T^{9} + 23616419385 T^{10} - 249612878 p T^{11} + 298984707 p^{2} T^{12} + 1951728 p^{3} T^{13} + 2644880 p^{4} T^{14} - 13312 p^{5} T^{15} + 19025 p^{6} T^{16} + 40 p^{7} T^{17} + 2 p^{9} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 46 T + 1292 T^{2} + 26530 T^{3} + 430839 T^{4} + 5783946 T^{5} + 66062922 T^{6} + 659196450 T^{7} + 5979674797 T^{8} + 52593327006 T^{9} + 491620931417 T^{10} + 52593327006 p T^{11} + 5979674797 p^{2} T^{12} + 659196450 p^{3} T^{13} + 66062922 p^{4} T^{14} + 5783946 p^{5} T^{15} + 430839 p^{6} T^{16} + 26530 p^{7} T^{17} + 1292 p^{8} T^{18} + 46 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.11055250262026344627040535938, −3.06114168490271076818956059545, −2.85817796444553647450494042365, −2.84481798507939631837339839205, −2.83237027677502976657695619190, −2.58614267779015558579085894071, −2.41527736164488166410623295936, −2.41270008271568103769160490613, −2.32359564807922738152504309809, −2.30524574483479622400760292523, −2.27780441248984615436577943997, −2.23688288488944499148457754468, −2.06266635010183292653945975867, −1.98609619005362648591970505689, −1.91278342165390681197190754143, −1.90384634501964224949512765236, −1.45156583586039181018672078601, −1.43182997044335718440897744201, −1.37130063040969910353596442411, −1.30379338924422612748500379674, −1.11754269100122193052911306484, −1.08432218026927522252175490172, −1.06856883179536101224375088155, −1.06496083818320876902815062921, −0.65409146398852029129338771939, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.65409146398852029129338771939, 1.06496083818320876902815062921, 1.06856883179536101224375088155, 1.08432218026927522252175490172, 1.11754269100122193052911306484, 1.30379338924422612748500379674, 1.37130063040969910353596442411, 1.43182997044335718440897744201, 1.45156583586039181018672078601, 1.90384634501964224949512765236, 1.91278342165390681197190754143, 1.98609619005362648591970505689, 2.06266635010183292653945975867, 2.23688288488944499148457754468, 2.27780441248984615436577943997, 2.30524574483479622400760292523, 2.32359564807922738152504309809, 2.41270008271568103769160490613, 2.41527736164488166410623295936, 2.58614267779015558579085894071, 2.83237027677502976657695619190, 2.84481798507939631837339839205, 2.85817796444553647450494042365, 3.06114168490271076818956059545, 3.11055250262026344627040535938
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.