Dirichlet series
| L(s) = 1 | + 2-s + 2·5-s + 2·10-s − 11·11-s + 13·13-s + 24·17-s − 14·19-s − 11·22-s + 10·23-s − 17·25-s + 13·26-s − 13·29-s + 8·31-s + 24·34-s − 13·37-s − 14·38-s + 10·41-s − 8·43-s + 10·46-s + 8·47-s + 18·49-s − 17·50-s + 35·53-s − 22·55-s − 13·58-s − 37·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.894·5-s + 0.632·10-s − 3.31·11-s + 3.60·13-s + 5.82·17-s − 3.21·19-s − 2.34·22-s + 2.08·23-s − 3.39·25-s + 2.54·26-s − 2.41·29-s + 1.43·31-s + 4.11·34-s − 2.13·37-s − 2.27·38-s + 1.56·41-s − 1.21·43-s + 1.47·46-s + 1.16·47-s + 18/7·49-s − 2.40·50-s + 4.80·53-s − 2.96·55-s − 1.70·58-s − 4.81·59-s − 0.256·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{20} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(155874.\) |
| Root analytic conductor: | \(1.81818\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9719337262\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9719337262\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 3 | \( 1 \) | |
| 23 | \( 1 - 10 T + 34 T^{2} - 131 T^{3} + 1442 T^{4} - 9767 T^{5} + 1442 p T^{6} - 131 p^{2} T^{7} + 34 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 5 | \( 1 - 2 T + 21 T^{2} - 32 T^{3} + 234 T^{4} - 264 T^{5} + 1756 T^{6} - 1422 T^{7} + 10432 T^{8} - 6384 T^{9} + 54361 T^{10} - 6384 p T^{11} + 10432 p^{2} T^{12} - 1422 p^{3} T^{13} + 1756 p^{4} T^{14} - 264 p^{5} T^{15} + 234 p^{6} T^{16} - 32 p^{7} T^{17} + 21 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 18 T^{2} - 22 T^{3} + 170 T^{4} + 176 T^{5} - 1201 T^{6} - 66 T^{7} + 13808 T^{8} - 4378 T^{9} - 119503 T^{10} - 4378 p T^{11} + 13808 p^{2} T^{12} - 66 p^{3} T^{13} - 1201 p^{4} T^{14} + 176 p^{5} T^{15} + 170 p^{6} T^{16} - 22 p^{7} T^{17} - 18 p^{8} T^{18} + p^{10} T^{20} \) | |
| 11 | \( 1 + p T + 4 p T^{2} + 4 p T^{3} - 36 p T^{4} - 218 p T^{5} - 50 p^{2} T^{6} - 28 p^{2} T^{7} + 259 p^{2} T^{8} + 939 p^{2} T^{9} + 2124 p^{2} T^{10} + 939 p^{3} T^{11} + 259 p^{4} T^{12} - 28 p^{5} T^{13} - 50 p^{6} T^{14} - 218 p^{6} T^{15} - 36 p^{7} T^{16} + 4 p^{8} T^{17} + 4 p^{9} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - p T + 68 T^{2} - 121 T^{3} - 873 T^{4} + 8390 T^{5} - 34053 T^{6} + 62579 T^{7} + 131840 T^{8} - 1594427 T^{9} + 7294143 T^{10} - 1594427 p T^{11} + 131840 p^{2} T^{12} + 62579 p^{3} T^{13} - 34053 p^{4} T^{14} + 8390 p^{5} T^{15} - 873 p^{6} T^{16} - 121 p^{7} T^{17} + 68 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 24 T + 262 T^{2} - 1766 T^{3} + 8263 T^{4} - 25873 T^{5} + 27963 T^{6} + 254049 T^{7} - 2176089 T^{8} + 11542392 T^{9} - 51031531 T^{10} + 11542392 p T^{11} - 2176089 p^{2} T^{12} + 254049 p^{3} T^{13} + 27963 p^{4} T^{14} - 25873 p^{5} T^{15} + 8263 p^{6} T^{16} - 1766 p^{7} T^{17} + 262 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 14 T + 45 T^{2} - 252 T^{3} - 1776 T^{4} + 1462 T^{5} + 36590 T^{6} + 45076 T^{7} - 361828 T^{8} + 215942 T^{9} + 9800889 T^{10} + 215942 p T^{11} - 361828 p^{2} T^{12} + 45076 p^{3} T^{13} + 36590 p^{4} T^{14} + 1462 p^{5} T^{15} - 1776 p^{6} T^{16} - 252 p^{7} T^{17} + 45 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 13 T + 41 T^{2} - 240 T^{3} - 1405 T^{4} + 9419 T^{5} + 62410 T^{6} - 364096 T^{7} - 3635431 T^{8} + 1467312 T^{9} + 99241429 T^{10} + 1467312 p T^{11} - 3635431 p^{2} T^{12} - 364096 p^{3} T^{13} + 62410 p^{4} T^{14} + 9419 p^{5} T^{15} - 1405 p^{6} T^{16} - 240 p^{7} T^{17} + 41 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 8 T - 77 T^{2} + 974 T^{3} + 1492 T^{4} - 56584 T^{5} + 107594 T^{6} + 1874024 T^{7} - 9959136 T^{8} - 25696906 T^{9} + 410937273 T^{10} - 25696906 p T^{11} - 9959136 p^{2} T^{12} + 1874024 p^{3} T^{13} + 107594 p^{4} T^{14} - 56584 p^{5} T^{15} + 1492 p^{6} T^{16} + 974 p^{7} T^{17} - 77 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 13 T + 33 T^{2} - 657 T^{3} - 4130 T^{4} + 16390 T^{5} + 255924 T^{6} + 349433 T^{7} - 4984034 T^{8} - 10629229 T^{9} + 104038681 T^{10} - 10629229 p T^{11} - 4984034 p^{2} T^{12} + 349433 p^{3} T^{13} + 255924 p^{4} T^{14} + 16390 p^{5} T^{15} - 4130 p^{6} T^{16} - 657 p^{7} T^{17} + 33 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 10 T + 136 T^{2} - 1643 T^{3} + 16123 T^{4} - 150363 T^{5} + 1259619 T^{6} - 10164498 T^{7} + 75970831 T^{8} - 528150586 T^{9} + 3482845541 T^{10} - 528150586 p T^{11} + 75970831 p^{2} T^{12} - 10164498 p^{3} T^{13} + 1259619 p^{4} T^{14} - 150363 p^{5} T^{15} + 16123 p^{6} T^{16} - 1643 p^{7} T^{17} + 136 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 8 T - 12 T^{2} - 561 T^{3} - 3609 T^{4} - 4485 T^{5} + 139195 T^{6} + 1135662 T^{7} + 3523939 T^{8} - 23870432 T^{9} - 324087611 T^{10} - 23870432 p T^{11} + 3523939 p^{2} T^{12} + 1135662 p^{3} T^{13} + 139195 p^{4} T^{14} - 4485 p^{5} T^{15} - 3609 p^{6} T^{16} - 561 p^{7} T^{17} - 12 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 - 4 T + 138 T^{2} - 670 T^{3} + 11095 T^{4} - 40491 T^{5} + 11095 p T^{6} - 670 p^{2} T^{7} + 138 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 - 35 T + 534 T^{2} - 4691 T^{3} + 30063 T^{4} - 190332 T^{5} + 837769 T^{6} + 5213181 T^{7} - 124869332 T^{8} + 1080670107 T^{9} - 7457270765 T^{10} + 1080670107 p T^{11} - 124869332 p^{2} T^{12} + 5213181 p^{3} T^{13} + 837769 p^{4} T^{14} - 190332 p^{5} T^{15} + 30063 p^{6} T^{16} - 4691 p^{7} T^{17} + 534 p^{8} T^{18} - 35 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 37 T + 650 T^{2} + 6918 T^{3} + 45103 T^{4} + 118937 T^{5} - 790848 T^{6} - 6814576 T^{7} + 912696 p T^{8} + 1457897138 T^{9} + 14750071633 T^{10} + 1457897138 p T^{11} + 912696 p^{3} T^{12} - 6814576 p^{3} T^{13} - 790848 p^{4} T^{14} + 118937 p^{5} T^{15} + 45103 p^{6} T^{16} + 6918 p^{7} T^{17} + 650 p^{8} T^{18} + 37 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 2 T + 86 T^{2} + 50 T^{3} + 4908 T^{4} - 31228 T^{5} + 212741 T^{6} - 1310478 T^{7} + 12234758 T^{8} - 31646784 T^{9} + 1455356297 T^{10} - 31646784 p T^{11} + 12234758 p^{2} T^{12} - 1310478 p^{3} T^{13} + 212741 p^{4} T^{14} - 31228 p^{5} T^{15} + 4908 p^{6} T^{16} + 50 p^{7} T^{17} + 86 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 14 T + 195 T^{2} - 3354 T^{3} + 39633 T^{4} - 419277 T^{5} + 4814065 T^{6} - 47450510 T^{7} + 426590845 T^{8} - 3846869807 T^{9} + 33199889581 T^{10} - 3846869807 p T^{11} + 426590845 p^{2} T^{12} - 47450510 p^{3} T^{13} + 4814065 p^{4} T^{14} - 419277 p^{5} T^{15} + 39633 p^{6} T^{16} - 3354 p^{7} T^{17} + 195 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 44 T + 897 T^{2} + 12210 T^{3} + 143289 T^{4} + 1622126 T^{5} + 16400193 T^{6} + 139879762 T^{7} + 1112568279 T^{8} + 9545296970 T^{9} + 83433721007 T^{10} + 9545296970 p T^{11} + 1112568279 p^{2} T^{12} + 139879762 p^{3} T^{13} + 16400193 p^{4} T^{14} + 1622126 p^{5} T^{15} + 143289 p^{6} T^{16} + 12210 p^{7} T^{17} + 897 p^{8} T^{18} + 44 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 49 T + 1272 T^{2} + 23111 T^{3} + 334395 T^{4} + 4161110 T^{5} + 46769369 T^{6} + 488164589 T^{7} + 4788201674 T^{8} + 44338020863 T^{9} + 388881210783 T^{10} + 44338020863 p T^{11} + 4788201674 p^{2} T^{12} + 488164589 p^{3} T^{13} + 46769369 p^{4} T^{14} + 4161110 p^{5} T^{15} + 334395 p^{6} T^{16} + 23111 p^{7} T^{17} + 1272 p^{8} T^{18} + 49 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 8 T + 227 T^{2} + 282 T^{3} + 24902 T^{4} - 31358 T^{5} + 2808680 T^{6} - 9747498 T^{7} + 221285300 T^{8} - 1405676580 T^{9} + 17486359837 T^{10} - 1405676580 p T^{11} + 221285300 p^{2} T^{12} - 9747498 p^{3} T^{13} + 2808680 p^{4} T^{14} - 31358 p^{5} T^{15} + 24902 p^{6} T^{16} + 282 p^{7} T^{17} + 227 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 17 T + 52 T^{2} + 406 T^{3} + 244 T^{4} - 1029 T^{5} - 761770 T^{6} + 8004680 T^{7} + 16074276 T^{8} - 470096066 T^{9} + 2367060235 T^{10} - 470096066 p T^{11} + 16074276 p^{2} T^{12} + 8004680 p^{3} T^{13} - 761770 p^{4} T^{14} - 1029 p^{5} T^{15} + 244 p^{6} T^{16} + 406 p^{7} T^{17} + 52 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 59 T + 1742 T^{2} + 36334 T^{3} + 614746 T^{4} + 8918095 T^{5} + 114846706 T^{6} + 1347527760 T^{7} + 14656918836 T^{8} + 150039947354 T^{9} + 1454906272975 T^{10} + 150039947354 p T^{11} + 14656918836 p^{2} T^{12} + 1347527760 p^{3} T^{13} + 114846706 p^{4} T^{14} + 8918095 p^{5} T^{15} + 614746 p^{6} T^{16} + 36334 p^{7} T^{17} + 1742 p^{8} T^{18} + 59 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 21 T + 300 T^{2} + 5341 T^{3} + 48477 T^{4} + 384748 T^{5} + 3678795 T^{6} - 8287925 T^{7} - 183989272 T^{8} - 2558529435 T^{9} - 54591704045 T^{10} - 2558529435 p T^{11} - 183989272 p^{2} T^{12} - 8287925 p^{3} T^{13} + 3678795 p^{4} T^{14} + 384748 p^{5} T^{15} + 48477 p^{6} T^{16} + 5341 p^{7} T^{17} + 300 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.24175354224245141625658363661, −4.11286471863170569367048338288, −4.06398097009918894390636376329, −4.05833156384700370280458848116, −3.80663306102771757322678215173, −3.77416494499585712612279610399, −3.28029255287437983328905322736, −3.27918015278216253357240241208, −3.17602857170913053762281366897, −3.09513770349596693308740979805, −3.07797937497947343737304953148, −2.99582754847970991752004440852, −2.95627273224154019564567615930, −2.70068233994524138417519893791, −2.29650531618561966296893804459, −2.22609868106672032291541574105, −2.11936519897292605700550337341, −1.91858013311585030217426989859, −1.71999146953551942794368333543, −1.57617710299954473505568307594, −1.44515293200066080233180991123, −1.22596095804089145143305794996, −1.02364203587348378176473845720, −0.76577126864484413450757757002, −0.12071622720488404338253531590, 0.12071622720488404338253531590, 0.76577126864484413450757757002, 1.02364203587348378176473845720, 1.22596095804089145143305794996, 1.44515293200066080233180991123, 1.57617710299954473505568307594, 1.71999146953551942794368333543, 1.91858013311585030217426989859, 2.11936519897292605700550337341, 2.22609868106672032291541574105, 2.29650531618561966296893804459, 2.70068233994524138417519893791, 2.95627273224154019564567615930, 2.99582754847970991752004440852, 3.07797937497947343737304953148, 3.09513770349596693308740979805, 3.17602857170913053762281366897, 3.27918015278216253357240241208, 3.28029255287437983328905322736, 3.77416494499585712612279610399, 3.80663306102771757322678215173, 4.05833156384700370280458848116, 4.06398097009918894390636376329, 4.11286471863170569367048338288, 4.24175354224245141625658363661
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.