Dirichlet series
| L(s) = 1 | + 3·2-s − 4-s + 10·5-s + 7-s − 11·8-s + 30·10-s − 13·13-s + 3·14-s − 9·16-s + 22·17-s − 2·19-s − 10·20-s − 10·23-s + 30·25-s − 39·26-s − 28-s − 10·29-s − 22·31-s + 7·32-s + 66·34-s + 10·35-s − 9·37-s − 6·38-s − 110·40-s + 25·41-s + 3·43-s − 30·46-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1/2·4-s + 4.47·5-s + 0.377·7-s − 3.88·8-s + 9.48·10-s − 3.60·13-s + 0.801·14-s − 9/4·16-s + 5.33·17-s − 0.458·19-s − 2.23·20-s − 2.08·23-s + 6·25-s − 7.64·26-s − 0.188·28-s − 1.85·29-s − 3.95·31-s + 1.23·32-s + 11.3·34-s + 1.69·35-s − 1.47·37-s − 0.973·38-s − 17.3·40-s + 3.90·41-s + 0.457·43-s − 4.42·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 23^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 23^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 23^{10} \cdot 29^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.40404\times 10^{16}\) |
| Root analytic conductor: | \(6.92345\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 23^{10} \cdot 29^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(38.18616079\) |
| \(L(\frac12)\) | \(\approx\) | \(38.18616079\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 23 | \( ( 1 + T )^{10} \) | |
| 29 | \( ( 1 + T )^{10} \) | |
| good | 2 | \( 1 - 3 T + 5 p T^{2} - 11 p T^{3} + 13 p^{2} T^{4} - 51 p T^{5} + 195 T^{6} - 163 p T^{7} + 135 p^{2} T^{8} - 817 T^{9} + 1219 T^{10} - 817 p T^{11} + 135 p^{4} T^{12} - 163 p^{4} T^{13} + 195 p^{4} T^{14} - 51 p^{6} T^{15} + 13 p^{8} T^{16} - 11 p^{8} T^{17} + 5 p^{9} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - 2 p T + 14 p T^{2} - 368 T^{3} + 323 p T^{4} - 6157 T^{5} + 20809 T^{6} - 63509 T^{7} + 176219 T^{8} - 447408 T^{9} + 1043891 T^{10} - 447408 p T^{11} + 176219 p^{2} T^{12} - 63509 p^{3} T^{13} + 20809 p^{4} T^{14} - 6157 p^{5} T^{15} + 323 p^{7} T^{16} - 368 p^{7} T^{17} + 14 p^{9} T^{18} - 2 p^{10} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - T + 27 T^{2} - 51 T^{3} + 320 T^{4} - 865 T^{5} + 2581 T^{6} - 7090 T^{7} + 19668 T^{8} - 37113 T^{9} + 146057 T^{10} - 37113 p T^{11} + 19668 p^{2} T^{12} - 7090 p^{3} T^{13} + 2581 p^{4} T^{14} - 865 p^{5} T^{15} + 320 p^{6} T^{16} - 51 p^{7} T^{17} + 27 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 46 T^{2} + 69 T^{3} + 1079 T^{4} + 266 p T^{5} + 19000 T^{6} + 5877 p T^{7} + 270077 T^{8} + 988343 T^{9} + 3200721 T^{10} + 988343 p T^{11} + 270077 p^{2} T^{12} + 5877 p^{4} T^{13} + 19000 p^{4} T^{14} + 266 p^{6} T^{15} + 1079 p^{6} T^{16} + 69 p^{7} T^{17} + 46 p^{8} T^{18} + p^{10} T^{20} \) | |
| 13 | \( 1 + p T + 145 T^{2} + 1138 T^{3} + 7974 T^{4} + 46970 T^{5} + 253908 T^{6} + 1216997 T^{7} + 5428597 T^{8} + 21924657 T^{9} + 82839053 T^{10} + 21924657 p T^{11} + 5428597 p^{2} T^{12} + 1216997 p^{3} T^{13} + 253908 p^{4} T^{14} + 46970 p^{5} T^{15} + 7974 p^{6} T^{16} + 1138 p^{7} T^{17} + 145 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 22 T + 279 T^{2} - 2560 T^{3} + 18885 T^{4} - 118890 T^{5} + 669412 T^{6} - 3469737 T^{7} + 16851864 T^{8} - 4518990 p T^{9} + 327849375 T^{10} - 4518990 p^{2} T^{11} + 16851864 p^{2} T^{12} - 3469737 p^{3} T^{13} + 669412 p^{4} T^{14} - 118890 p^{5} T^{15} + 18885 p^{6} T^{16} - 2560 p^{7} T^{17} + 279 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 2 T + 5 p T^{2} + 311 T^{3} + 4337 T^{4} + 18975 T^{5} + 138780 T^{6} + 661855 T^{7} + 3627427 T^{8} + 16047275 T^{9} + 77334365 T^{10} + 16047275 p T^{11} + 3627427 p^{2} T^{12} + 661855 p^{3} T^{13} + 138780 p^{4} T^{14} + 18975 p^{5} T^{15} + 4337 p^{6} T^{16} + 311 p^{7} T^{17} + 5 p^{9} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 22 T + 430 T^{2} + 5551 T^{3} + 65126 T^{4} + 617363 T^{5} + 5422722 T^{6} + 41104187 T^{7} + 291555454 T^{8} + 1830808995 T^{9} + 10796917527 T^{10} + 1830808995 p T^{11} + 291555454 p^{2} T^{12} + 41104187 p^{3} T^{13} + 5422722 p^{4} T^{14} + 617363 p^{5} T^{15} + 65126 p^{6} T^{16} + 5551 p^{7} T^{17} + 430 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 9 T + 272 T^{2} + 2244 T^{3} + 36506 T^{4} + 266344 T^{5} + 3108815 T^{6} + 19865478 T^{7} + 184485854 T^{8} + 1025174520 T^{9} + 7957552221 T^{10} + 1025174520 p T^{11} + 184485854 p^{2} T^{12} + 19865478 p^{3} T^{13} + 3108815 p^{4} T^{14} + 266344 p^{5} T^{15} + 36506 p^{6} T^{16} + 2244 p^{7} T^{17} + 272 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 25 T + 495 T^{2} - 6723 T^{3} + 81455 T^{4} - 815800 T^{5} + 7630229 T^{6} - 62837308 T^{7} + 492494695 T^{8} - 3473801813 T^{9} + 23398531031 T^{10} - 3473801813 p T^{11} + 492494695 p^{2} T^{12} - 62837308 p^{3} T^{13} + 7630229 p^{4} T^{14} - 815800 p^{5} T^{15} + 81455 p^{6} T^{16} - 6723 p^{7} T^{17} + 495 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 3 T + 247 T^{2} - 665 T^{3} + 31054 T^{4} - 74510 T^{5} + 2630158 T^{6} - 5674198 T^{7} + 165896276 T^{8} - 321130465 T^{9} + 8083839367 T^{10} - 321130465 p T^{11} + 165896276 p^{2} T^{12} - 5674198 p^{3} T^{13} + 2630158 p^{4} T^{14} - 74510 p^{5} T^{15} + 31054 p^{6} T^{16} - 665 p^{7} T^{17} + 247 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 17 T + 313 T^{2} - 2557 T^{3} + 25054 T^{4} - 85032 T^{5} + 579788 T^{6} + 4662563 T^{7} - 30698536 T^{8} + 639621255 T^{9} - 3017746273 T^{10} + 639621255 p T^{11} - 30698536 p^{2} T^{12} + 4662563 p^{3} T^{13} + 579788 p^{4} T^{14} - 85032 p^{5} T^{15} + 25054 p^{6} T^{16} - 2557 p^{7} T^{17} + 313 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 43 T + 1108 T^{2} - 20557 T^{3} + 306875 T^{4} - 3849325 T^{5} + 42092239 T^{6} - 409427041 T^{7} + 3616962404 T^{8} - 29372469128 T^{9} + 221665477065 T^{10} - 29372469128 p T^{11} + 3616962404 p^{2} T^{12} - 409427041 p^{3} T^{13} + 42092239 p^{4} T^{14} - 3849325 p^{5} T^{15} + 306875 p^{6} T^{16} - 20557 p^{7} T^{17} + 1108 p^{8} T^{18} - 43 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 7 T + 368 T^{2} - 2593 T^{3} + 66263 T^{4} - 460573 T^{5} + 7816897 T^{6} - 52222655 T^{7} + 676063465 T^{8} - 4192119863 T^{9} + 45083176781 T^{10} - 4192119863 p T^{11} + 676063465 p^{2} T^{12} - 52222655 p^{3} T^{13} + 7816897 p^{4} T^{14} - 460573 p^{5} T^{15} + 66263 p^{6} T^{16} - 2593 p^{7} T^{17} + 368 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 6 T + 442 T^{2} + 2408 T^{3} + 93753 T^{4} + 455424 T^{5} + 12598899 T^{6} + 54110890 T^{7} + 1194644788 T^{8} + 4506335398 T^{9} + 83995741755 T^{10} + 4506335398 p T^{11} + 1194644788 p^{2} T^{12} + 54110890 p^{3} T^{13} + 12598899 p^{4} T^{14} + 455424 p^{5} T^{15} + 93753 p^{6} T^{16} + 2408 p^{7} T^{17} + 442 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 11 T + 440 T^{2} - 3258 T^{3} + 82372 T^{4} - 437749 T^{5} + 148236 p T^{6} - 43101729 T^{7} + 945457911 T^{8} - 3694828149 T^{9} + 72034637003 T^{10} - 3694828149 p T^{11} + 945457911 p^{2} T^{12} - 43101729 p^{3} T^{13} + 148236 p^{5} T^{14} - 437749 p^{5} T^{15} + 82372 p^{6} T^{16} - 3258 p^{7} T^{17} + 440 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 17 T + 487 T^{2} - 5429 T^{3} + 95541 T^{4} - 839054 T^{5} + 12202961 T^{6} - 93606104 T^{7} + 1206692029 T^{8} - 8257557627 T^{9} + 95318141337 T^{10} - 8257557627 p T^{11} + 1206692029 p^{2} T^{12} - 93606104 p^{3} T^{13} + 12202961 p^{4} T^{14} - 839054 p^{5} T^{15} + 95541 p^{6} T^{16} - 5429 p^{7} T^{17} + 487 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 44 T + 1512 T^{2} + 35899 T^{3} + 726529 T^{4} + 12037217 T^{5} + 176244267 T^{6} + 2227972384 T^{7} + 25319414428 T^{8} + 253707899388 T^{9} + 2301758109781 T^{10} + 253707899388 p T^{11} + 25319414428 p^{2} T^{12} + 2227972384 p^{3} T^{13} + 176244267 p^{4} T^{14} + 12037217 p^{5} T^{15} + 726529 p^{6} T^{16} + 35899 p^{7} T^{17} + 1512 p^{8} T^{18} + 44 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 5 T + 453 T^{2} - 1654 T^{3} + 91613 T^{4} - 178830 T^{5} + 10951097 T^{6} + 1909376 T^{7} + 919392184 T^{8} + 2065356423 T^{9} + 69768644457 T^{10} + 2065356423 p T^{11} + 919392184 p^{2} T^{12} + 1909376 p^{3} T^{13} + 10951097 p^{4} T^{14} - 178830 p^{5} T^{15} + 91613 p^{6} T^{16} - 1654 p^{7} T^{17} + 453 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 32 T + 1006 T^{2} - 20223 T^{3} + 380602 T^{4} - 5730612 T^{5} + 80923753 T^{6} - 985170346 T^{7} + 11305014451 T^{8} - 115208135432 T^{9} + 1109307126255 T^{10} - 115208135432 p T^{11} + 11305014451 p^{2} T^{12} - 985170346 p^{3} T^{13} + 80923753 p^{4} T^{14} - 5730612 p^{5} T^{15} + 380602 p^{6} T^{16} - 20223 p^{7} T^{17} + 1006 p^{8} T^{18} - 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 10 T + 554 T^{2} - 5842 T^{3} + 155645 T^{4} - 1640385 T^{5} + 28935836 T^{6} - 290928628 T^{7} + 3903118975 T^{8} - 35927833885 T^{9} + 397465410729 T^{10} - 35927833885 p T^{11} + 3903118975 p^{2} T^{12} - 290928628 p^{3} T^{13} + 28935836 p^{4} T^{14} - 1640385 p^{5} T^{15} + 155645 p^{6} T^{16} - 5842 p^{7} T^{17} + 554 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 6 T + 456 T^{2} - 3944 T^{3} + 108368 T^{4} - 1175483 T^{5} + 18618490 T^{6} - 212770829 T^{7} + 2555320728 T^{8} - 27177265474 T^{9} + 279700991985 T^{10} - 27177265474 p T^{11} + 2555320728 p^{2} T^{12} - 212770829 p^{3} T^{13} + 18618490 p^{4} T^{14} - 1175483 p^{5} T^{15} + 108368 p^{6} T^{16} - 3944 p^{7} T^{17} + 456 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.64954921273297616048125137165, −2.55509307355134718495962779174, −2.51205465427911095771362343415, −2.39712948518771264915961787827, −2.30915054002250223885310225836, −2.29230240884898125424820339851, −2.19942696842313804537932268815, −2.15083651308674827420570360807, −2.01685752022652553537110085431, −1.85910698654649440966949804915, −1.80959912890504346431359868648, −1.78293752782965660662000653590, −1.78048729446068969283411079830, −1.69803498063745760269129719588, −1.63588504466586665174512503449, −1.54561915043543294176945714411, −1.13748789484607988693499755703, −1.03603234036042559779668220424, −0.75371198959746194983639070131, −0.75324726970035300658404284586, −0.65640124901644462917629175007, −0.62058253710722327872943916501, −0.58023921320725804945847735359, −0.39075256170281089234378915843, −0.10601654928492792570679309918, 0.10601654928492792570679309918, 0.39075256170281089234378915843, 0.58023921320725804945847735359, 0.62058253710722327872943916501, 0.65640124901644462917629175007, 0.75324726970035300658404284586, 0.75371198959746194983639070131, 1.03603234036042559779668220424, 1.13748789484607988693499755703, 1.54561915043543294176945714411, 1.63588504466586665174512503449, 1.69803498063745760269129719588, 1.78048729446068969283411079830, 1.78293752782965660662000653590, 1.80959912890504346431359868648, 1.85910698654649440966949804915, 2.01685752022652553537110085431, 2.15083651308674827420570360807, 2.19942696842313804537932268815, 2.29230240884898125424820339851, 2.30915054002250223885310225836, 2.39712948518771264915961787827, 2.51205465427911095771362343415, 2.55509307355134718495962779174, 2.64954921273297616048125137165
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.