Dirichlet series
| L(s) = 1 | − 2·2-s + 4·4-s + 10·5-s − 7-s − 4·8-s − 20·10-s − 8·11-s + 13-s + 2·14-s + 6·16-s + 4·19-s + 40·20-s + 16·22-s + 6·23-s + 55·25-s − 2·26-s − 4·28-s − 16·29-s + 18·31-s − 6·32-s − 10·35-s + 4·37-s − 8·38-s − 40·40-s − 6·41-s − 15·43-s − 32·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2·4-s + 4.47·5-s − 0.377·7-s − 1.41·8-s − 6.32·10-s − 2.41·11-s + 0.277·13-s + 0.534·14-s + 3/2·16-s + 0.917·19-s + 8.94·20-s + 3.41·22-s + 1.25·23-s + 11·25-s − 0.392·26-s − 0.755·28-s − 2.97·29-s + 3.23·31-s − 1.06·32-s − 1.69·35-s + 0.657·37-s − 1.29·38-s − 6.32·40-s − 0.937·41-s − 2.28·43-s − 4.82·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 5^{10} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.94686\times 10^{6}\) |
| Root analytic conductor: | \(2.16130\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 5^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.721944391\) |
| \(L(\frac12)\) | \(\approx\) | \(6.721944391\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - T )^{10} \) | |
| 13 | \( 1 - T - 21 T^{2} + 106 T^{3} + 185 T^{4} - 2013 T^{5} + 185 p T^{6} + 106 p^{2} T^{7} - 21 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 + p T - p^{2} T^{3} - 3 p T^{4} - p T^{5} + p^{3} T^{6} + 3 p^{2} T^{7} - p^{2} T^{9} + 3 p^{2} T^{10} - p^{3} T^{11} + 3 p^{5} T^{13} + p^{7} T^{14} - p^{6} T^{15} - 3 p^{7} T^{16} - p^{9} T^{17} + p^{10} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 + T - 12 T^{2} + T^{3} + 88 T^{4} - 37 T^{5} + 76 T^{6} + 359 T^{7} - 5281 T^{8} - 4 p^{2} T^{9} + 8116 p T^{10} - 4 p^{3} T^{11} - 5281 p^{2} T^{12} + 359 p^{3} T^{13} + 76 p^{4} T^{14} - 37 p^{5} T^{15} + 88 p^{6} T^{16} + p^{7} T^{17} - 12 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 8 T + 9 T^{2} - 136 T^{3} - 537 T^{4} + 136 T^{5} + 6038 T^{6} + 21216 T^{7} + 31725 T^{8} - 137272 T^{9} - 918177 T^{10} - 137272 p T^{11} + 31725 p^{2} T^{12} + 21216 p^{3} T^{13} + 6038 p^{4} T^{14} + 136 p^{5} T^{15} - 537 p^{6} T^{16} - 136 p^{7} T^{17} + 9 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 35 T^{2} - 28 T^{3} + 315 T^{4} + 324 T^{5} - 42 p T^{6} + 28406 T^{7} + 117019 T^{8} - 442610 T^{9} - 3489795 T^{10} - 442610 p T^{11} + 117019 p^{2} T^{12} + 28406 p^{3} T^{13} - 42 p^{5} T^{14} + 324 p^{5} T^{15} + 315 p^{6} T^{16} - 28 p^{7} T^{17} - 35 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 - 4 T - 71 T^{2} + 188 T^{3} + 3347 T^{4} - 5424 T^{5} - 5882 p T^{6} + 91244 T^{7} + 2934257 T^{8} - 702040 T^{9} - 61778793 T^{10} - 702040 p T^{11} + 2934257 p^{2} T^{12} + 91244 p^{3} T^{13} - 5882 p^{5} T^{14} - 5424 p^{5} T^{15} + 3347 p^{6} T^{16} + 188 p^{7} T^{17} - 71 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 6 T - 59 T^{2} + 462 T^{3} + 1899 T^{4} - 17976 T^{5} - 35110 T^{6} + 381456 T^{7} + 619997 T^{8} - 3636078 T^{9} - 11442537 T^{10} - 3636078 p T^{11} + 619997 p^{2} T^{12} + 381456 p^{3} T^{13} - 35110 p^{4} T^{14} - 17976 p^{5} T^{15} + 1899 p^{6} T^{16} + 462 p^{7} T^{17} - 59 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 16 T + 53 T^{2} - 280 T^{3} - 3 p T^{4} + 14930 T^{5} + 13984 T^{6} + 100288 T^{7} + 3893633 T^{8} + 2571612 T^{9} - 117536991 T^{10} + 2571612 p T^{11} + 3893633 p^{2} T^{12} + 100288 p^{3} T^{13} + 13984 p^{4} T^{14} + 14930 p^{5} T^{15} - 3 p^{7} T^{16} - 280 p^{7} T^{17} + 53 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( ( 1 - 9 T + 125 T^{2} - 938 T^{3} + 7513 T^{4} - 40255 T^{5} + 7513 p T^{6} - 938 p^{2} T^{7} + 125 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 37 | \( 1 - 4 T - 105 T^{2} + 812 T^{3} + 5223 T^{4} - 60656 T^{5} - 48342 T^{6} + 2598544 T^{7} - 6880531 T^{8} - 41122020 T^{9} + 443418561 T^{10} - 41122020 p T^{11} - 6880531 p^{2} T^{12} + 2598544 p^{3} T^{13} - 48342 p^{4} T^{14} - 60656 p^{5} T^{15} + 5223 p^{6} T^{16} + 812 p^{7} T^{17} - 105 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 6 T - 71 T^{2} - 198 T^{3} + 2037 T^{4} - 11598 T^{5} - 152896 T^{6} - 303432 T^{7} + 7372433 T^{8} + 28857882 T^{9} - 198785607 T^{10} + 28857882 p T^{11} + 7372433 p^{2} T^{12} - 303432 p^{3} T^{13} - 152896 p^{4} T^{14} - 11598 p^{5} T^{15} + 2037 p^{6} T^{16} - 198 p^{7} T^{17} - 71 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 15 T - 20 T^{2} - 569 T^{3} + 8062 T^{4} + 40863 T^{5} - 572708 T^{6} - 1600281 T^{7} + 26293173 T^{8} - 7330552 T^{9} - 1511025092 T^{10} - 7330552 p T^{11} + 26293173 p^{2} T^{12} - 1600281 p^{3} T^{13} - 572708 p^{4} T^{14} + 40863 p^{5} T^{15} + 8062 p^{6} T^{16} - 569 p^{7} T^{17} - 20 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 - 10 T + 145 T^{2} - 24 p T^{3} + 10018 T^{4} - 69286 T^{5} + 10018 p T^{6} - 24 p^{3} T^{7} + 145 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( ( 1 + 20 T + 5 p T^{2} + 2368 T^{3} + 17722 T^{4} + 123096 T^{5} + 17722 p T^{6} + 2368 p^{2} T^{7} + 5 p^{4} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 59 | \( 1 + 12 T - 109 T^{2} - 1836 T^{3} + 7077 T^{4} + 158166 T^{5} - 285300 T^{6} - 9000432 T^{7} + 1945509 T^{8} + 240199128 T^{9} + 613215327 T^{10} + 240199128 p T^{11} + 1945509 p^{2} T^{12} - 9000432 p^{3} T^{13} - 285300 p^{4} T^{14} + 158166 p^{5} T^{15} + 7077 p^{6} T^{16} - 1836 p^{7} T^{17} - 109 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 11 T - 170 T^{2} - 1355 T^{3} + 25322 T^{4} + 107353 T^{5} - 2797432 T^{6} - 6232727 T^{7} + 234614309 T^{8} + 184344598 T^{9} - 15525735436 T^{10} + 184344598 p T^{11} + 234614309 p^{2} T^{12} - 6232727 p^{3} T^{13} - 2797432 p^{4} T^{14} + 107353 p^{5} T^{15} + 25322 p^{6} T^{16} - 1355 p^{7} T^{17} - 170 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 5 T - 266 T^{2} - 1217 T^{3} + 41876 T^{4} + 161677 T^{5} - 4557484 T^{6} - 11713703 T^{7} + 393911771 T^{8} + 360277294 T^{9} - 28368034144 T^{10} + 360277294 p T^{11} + 393911771 p^{2} T^{12} - 11713703 p^{3} T^{13} - 4557484 p^{4} T^{14} + 161677 p^{5} T^{15} + 41876 p^{6} T^{16} - 1217 p^{7} T^{17} - 266 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 10 T - 233 T^{2} - 1642 T^{3} + 40821 T^{4} + 175708 T^{5} - 4974832 T^{6} - 10835286 T^{7} + 479818097 T^{8} + 291547112 T^{9} - 37711050573 T^{10} + 291547112 p T^{11} + 479818097 p^{2} T^{12} - 10835286 p^{3} T^{13} - 4974832 p^{4} T^{14} + 175708 p^{5} T^{15} + 40821 p^{6} T^{16} - 1642 p^{7} T^{17} - 233 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( ( 1 - T + 129 T^{2} - 148 T^{3} + 13711 T^{4} - 1257 T^{5} + 13711 p T^{6} - 148 p^{2} T^{7} + 129 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( ( 1 + 17 T + 369 T^{2} + 4286 T^{3} + 54349 T^{4} + 467343 T^{5} + 54349 p T^{6} + 4286 p^{2} T^{7} + 369 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 83 | \( ( 1 - 16 T + 299 T^{2} - 3252 T^{3} + 38638 T^{4} - 325648 T^{5} + 38638 p T^{6} - 3252 p^{2} T^{7} + 299 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 + 4 T - 327 T^{2} - 1320 T^{3} + 60217 T^{4} + 221568 T^{5} - 7481300 T^{6} - 21676602 T^{7} + 726349165 T^{8} + 826863906 T^{9} - 65088711959 T^{10} + 826863906 p T^{11} + 726349165 p^{2} T^{12} - 21676602 p^{3} T^{13} - 7481300 p^{4} T^{14} + 221568 p^{5} T^{15} + 60217 p^{6} T^{16} - 1320 p^{7} T^{17} - 327 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 11 T - 234 T^{2} + 419 T^{3} + 54134 T^{4} + 86137 T^{5} - 5323566 T^{6} - 30313009 T^{7} + 338701565 T^{8} + 1490050320 T^{9} - 15821303860 T^{10} + 1490050320 p T^{11} + 338701565 p^{2} T^{12} - 30313009 p^{3} T^{13} - 5323566 p^{4} T^{14} + 86137 p^{5} T^{15} + 54134 p^{6} T^{16} + 419 p^{7} T^{17} - 234 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.89879430555165466937568999005, −3.78607418182554162967690422567, −3.58130681870773134895965362900, −3.40558793610226474532718095938, −3.39197185535506455895658665966, −3.38912327447566268442115279191, −3.24520787399199107961889446990, −2.91354768050988957628691420427, −2.76128578114223689235094385531, −2.75566300512355922143914187531, −2.73814304254478478460589027174, −2.69122156272706521870834381911, −2.55432526714454964029466166101, −2.34299227922456525278932285277, −2.07556207571406793965252471671, −1.91030158703125581405398973645, −1.81544647822290738232505139816, −1.74086571469785110810284388402, −1.74044873838575182690598360895, −1.56820864620310750549355133215, −1.30818553521700437974738275752, −1.09017330853391288163438432145, −0.917692887703527879561682108541, −0.69843594285636750727066279333, −0.28254792803385107052578559093, 0.28254792803385107052578559093, 0.69843594285636750727066279333, 0.917692887703527879561682108541, 1.09017330853391288163438432145, 1.30818553521700437974738275752, 1.56820864620310750549355133215, 1.74044873838575182690598360895, 1.74086571469785110810284388402, 1.81544647822290738232505139816, 1.91030158703125581405398973645, 2.07556207571406793965252471671, 2.34299227922456525278932285277, 2.55432526714454964029466166101, 2.69122156272706521870834381911, 2.73814304254478478460589027174, 2.75566300512355922143914187531, 2.76128578114223689235094385531, 2.91354768050988957628691420427, 3.24520787399199107961889446990, 3.38912327447566268442115279191, 3.39197185535506455895658665966, 3.40558793610226474532718095938, 3.58130681870773134895965362900, 3.78607418182554162967690422567, 3.89879430555165466937568999005
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.