Dirichlet series
| L(s) = 1 | − 2·2-s + 5·3-s + 2·4-s + 4·5-s − 10·6-s − 7-s + 10·9-s − 8·10-s − 5·11-s + 10·12-s + 10·13-s + 2·14-s + 20·15-s − 16-s − 2·17-s − 20·18-s + 3·19-s + 8·20-s − 5·21-s + 10·22-s − 16·23-s + 17·25-s − 20·26-s + 5·27-s − 2·28-s − 40·30-s − 5·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.88·3-s + 4-s + 1.78·5-s − 4.08·6-s − 0.377·7-s + 10/3·9-s − 2.52·10-s − 1.50·11-s + 2.88·12-s + 2.77·13-s + 0.534·14-s + 5.16·15-s − 1/4·16-s − 0.485·17-s − 4.71·18-s + 0.688·19-s + 1.78·20-s − 1.09·21-s + 2.13·22-s − 3.33·23-s + 17/5·25-s − 3.92·26-s + 0.962·27-s − 0.377·28-s − 7.30·30-s − 0.898·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{10} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{10} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 7^{10} \cdot 11^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(455.922\) |
| Root analytic conductor: | \(1.35814\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 7^{10} \cdot 11^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.603221619\) |
| \(L(\frac12)\) | \(\approx\) | \(1.603221619\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T + T^{2} )^{5} \) |
| 7 | \( 1 + T - 15 T^{2} - 36 T^{3} + 60 T^{4} + 402 T^{5} + 60 p T^{6} - 36 p^{2} T^{7} - 15 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | |
| 11 | \( ( 1 + T + T^{2} )^{5} \) | |
| good | 2 | \( 1 + p T + p T^{2} - 3 T^{4} - 3 p T^{5} + 3 p T^{6} + 9 p T^{7} + 9 T^{8} - 17 p T^{9} - 31 p T^{10} - 17 p^{2} T^{11} + 9 p^{2} T^{12} + 9 p^{4} T^{13} + 3 p^{5} T^{14} - 3 p^{6} T^{15} - 3 p^{6} T^{16} + p^{9} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - 4 T - T^{2} + 36 T^{3} - 69 T^{4} - 36 T^{5} + 246 T^{6} - 132 T^{7} - 279 T^{8} - 16 p^{2} T^{9} + 2929 T^{10} - 16 p^{3} T^{11} - 279 p^{2} T^{12} - 132 p^{3} T^{13} + 246 p^{4} T^{14} - 36 p^{5} T^{15} - 69 p^{6} T^{16} + 36 p^{7} T^{17} - p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( ( 1 - 5 T + 27 T^{2} - 180 T^{3} + 732 T^{4} - 2514 T^{5} + 732 p T^{6} - 180 p^{2} T^{7} + 27 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 17 | \( 1 + 2 T - 73 T^{2} - 90 T^{3} + 3192 T^{4} + 2424 T^{5} - 98319 T^{6} - 38142 T^{7} + 2348199 T^{8} + 280406 T^{9} - 44440892 T^{10} + 280406 p T^{11} + 2348199 p^{2} T^{12} - 38142 p^{3} T^{13} - 98319 p^{4} T^{14} + 2424 p^{5} T^{15} + 3192 p^{6} T^{16} - 90 p^{7} T^{17} - 73 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 3 T - 44 T^{2} + 81 T^{3} + 990 T^{4} - 603 T^{5} - 16476 T^{6} + 16119 T^{7} + 196221 T^{8} - 324198 T^{9} - 1776456 T^{10} - 324198 p T^{11} + 196221 p^{2} T^{12} + 16119 p^{3} T^{13} - 16476 p^{4} T^{14} - 603 p^{5} T^{15} + 990 p^{6} T^{16} + 81 p^{7} T^{17} - 44 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 16 T + 71 T^{2} - 72 T^{3} + 24 T^{4} + 14832 T^{5} + 74109 T^{6} + 71904 T^{7} + 239211 T^{8} + 5124568 T^{9} + 32231056 T^{10} + 5124568 p T^{11} + 239211 p^{2} T^{12} + 71904 p^{3} T^{13} + 74109 p^{4} T^{14} + 14832 p^{5} T^{15} + 24 p^{6} T^{16} - 72 p^{7} T^{17} + 71 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( ( 1 + 49 T^{2} + 24 T^{3} + 1765 T^{4} + 102 T^{5} + 1765 p T^{6} + 24 p^{2} T^{7} + 49 p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| 31 | \( 1 + 5 T - 122 T^{2} - 441 T^{3} + 9747 T^{4} + 24042 T^{5} - 546756 T^{6} - 692730 T^{7} + 24442245 T^{8} + 9807311 T^{9} - 844500050 T^{10} + 9807311 p T^{11} + 24442245 p^{2} T^{12} - 692730 p^{3} T^{13} - 546756 p^{4} T^{14} + 24042 p^{5} T^{15} + 9747 p^{6} T^{16} - 441 p^{7} T^{17} - 122 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 15 T + 10 T^{2} - 603 T^{3} + 1314 T^{4} + 25845 T^{5} - 126420 T^{6} - 536415 T^{7} + 9212421 T^{8} + 1856814 T^{9} - 461695068 T^{10} + 1856814 p T^{11} + 9212421 p^{2} T^{12} - 536415 p^{3} T^{13} - 126420 p^{4} T^{14} + 25845 p^{5} T^{15} + 1314 p^{6} T^{16} - 603 p^{7} T^{17} + 10 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( ( 1 + 22 T + 329 T^{2} + 3256 T^{3} + 27018 T^{4} + 181492 T^{5} + 27018 p T^{6} + 3256 p^{2} T^{7} + 329 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 3 T + 107 T^{2} - 114 T^{3} + 5299 T^{4} - 1683 T^{5} + 5299 p T^{6} - 114 p^{2} T^{7} + 107 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 - 2 T - 37 T^{2} - 354 T^{3} - 1920 T^{4} + 14694 T^{5} + 211017 T^{6} + 985758 T^{7} + 2945283 T^{8} - 50876312 T^{9} - 605046656 T^{10} - 50876312 p T^{11} + 2945283 p^{2} T^{12} + 985758 p^{3} T^{13} + 211017 p^{4} T^{14} + 14694 p^{5} T^{15} - 1920 p^{6} T^{16} - 354 p^{7} T^{17} - 37 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 6 T - 133 T^{2} - 354 T^{3} + 10587 T^{4} - 5544 T^{5} - 677754 T^{6} + 507696 T^{7} + 35054169 T^{8} + 3665190 T^{9} - 1656194019 T^{10} + 3665190 p T^{11} + 35054169 p^{2} T^{12} + 507696 p^{3} T^{13} - 677754 p^{4} T^{14} - 5544 p^{5} T^{15} + 10587 p^{6} T^{16} - 354 p^{7} T^{17} - 133 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 16 T - 109 T^{2} - 1800 T^{3} + 25080 T^{4} + 228672 T^{5} - 2449527 T^{6} - 10271040 T^{7} + 253584315 T^{8} + 417790408 T^{9} - 15321919856 T^{10} + 417790408 p T^{11} + 253584315 p^{2} T^{12} - 10271040 p^{3} T^{13} - 2449527 p^{4} T^{14} + 228672 p^{5} T^{15} + 25080 p^{6} T^{16} - 1800 p^{7} T^{17} - 109 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 12 T + 19 T^{2} - 900 T^{3} - 12441 T^{4} - 72288 T^{5} + 130650 T^{6} + 4443912 T^{7} + 34513281 T^{8} + 10060452 T^{9} - 1324169655 T^{10} + 10060452 p T^{11} + 34513281 p^{2} T^{12} + 4443912 p^{3} T^{13} + 130650 p^{4} T^{14} - 72288 p^{5} T^{15} - 12441 p^{6} T^{16} - 900 p^{7} T^{17} + 19 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 7 T - 194 T^{2} - 1203 T^{3} + 21423 T^{4} + 103134 T^{5} - 1828416 T^{6} - 6223746 T^{7} + 129539373 T^{8} + 171080449 T^{9} - 8649139910 T^{10} + 171080449 p T^{11} + 129539373 p^{2} T^{12} - 6223746 p^{3} T^{13} - 1828416 p^{4} T^{14} + 103134 p^{5} T^{15} + 21423 p^{6} T^{16} - 1203 p^{7} T^{17} - 194 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 - 24 T + 421 T^{2} - 5436 T^{3} + 57793 T^{4} - 524712 T^{5} + 57793 p T^{6} - 5436 p^{2} T^{7} + 421 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 + 17 T + 22 T^{2} + 51 T^{3} + 12429 T^{4} + 38334 T^{5} + 91686 T^{6} + 7567200 T^{7} + 6227061 T^{8} + 177295901 T^{9} + 7596523564 T^{10} + 177295901 p T^{11} + 6227061 p^{2} T^{12} + 7567200 p^{3} T^{13} + 91686 p^{4} T^{14} + 38334 p^{5} T^{15} + 12429 p^{6} T^{16} + 51 p^{7} T^{17} + 22 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 7 T - 92 T^{2} + 1035 T^{3} + 8313 T^{4} - 163986 T^{5} + 174174 T^{6} + 5014596 T^{7} - 64520235 T^{8} + 8158399 T^{9} + 3747754354 T^{10} + 8158399 p T^{11} - 64520235 p^{2} T^{12} + 5014596 p^{3} T^{13} + 174174 p^{4} T^{14} - 163986 p^{5} T^{15} + 8313 p^{6} T^{16} + 1035 p^{7} T^{17} - 92 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 + 12 T + 295 T^{2} + 3168 T^{3} + 40450 T^{4} + 363240 T^{5} + 40450 p T^{6} + 3168 p^{2} T^{7} + 295 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 - 6 T - 241 T^{2} + 978 T^{3} + 28551 T^{4} - 40212 T^{5} - 3189150 T^{6} + 2254020 T^{7} + 350296557 T^{8} - 203364690 T^{9} - 32844883695 T^{10} - 203364690 p T^{11} + 350296557 p^{2} T^{12} + 2254020 p^{3} T^{13} - 3189150 p^{4} T^{14} - 40212 p^{5} T^{15} + 28551 p^{6} T^{16} + 978 p^{7} T^{17} - 241 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 + 14 T + 159 T^{2} + 2856 T^{3} + 33045 T^{4} + 251706 T^{5} + 33045 p T^{6} + 2856 p^{2} T^{7} + 159 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−4.75237587861473807756639083254, −4.60145515814100237089663011469, −4.55197058577223775146162950986, −4.41962153051432399058732815301, −4.18329560277037879040960237098, −3.84629497207179862484380209848, −3.78739467726706002865070854040, −3.77902549510911065146059293834, −3.51042562456795163934746149939, −3.41080017339644472557022223623, −3.33911796372841935173982936034, −3.24761559956119449164551098898, −3.23111926566222503422773079308, −2.90730816725107387492205161598, −2.58014371197814903976251938384, −2.57364319730696088751972676224, −2.57347526539056940725661369465, −2.10053047807703309129117907907, −2.07739223728395573809985178677, −1.90692326552768847779218561111, −1.68196289510142251257437016136, −1.48008530645475742953934655230, −1.47775343198064143655500908774, −1.24676536414849647679813762863, −0.27680747400590555259169607297, 0.27680747400590555259169607297, 1.24676536414849647679813762863, 1.47775343198064143655500908774, 1.48008530645475742953934655230, 1.68196289510142251257437016136, 1.90692326552768847779218561111, 2.07739223728395573809985178677, 2.10053047807703309129117907907, 2.57347526539056940725661369465, 2.57364319730696088751972676224, 2.58014371197814903976251938384, 2.90730816725107387492205161598, 3.23111926566222503422773079308, 3.24761559956119449164551098898, 3.33911796372841935173982936034, 3.41080017339644472557022223623, 3.51042562456795163934746149939, 3.77902549510911065146059293834, 3.78739467726706002865070854040, 3.84629497207179862484380209848, 4.18329560277037879040960237098, 4.41962153051432399058732815301, 4.55197058577223775146162950986, 4.60145515814100237089663011469, 4.75237587861473807756639083254
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.