Dirichlet series
| L(s) = 1 | + 10·3-s + 2·5-s + 11·7-s + 55·9-s + 11·11-s + 20·15-s + 13·17-s + 18·19-s + 110·21-s − 18·25-s + 220·27-s + 5·29-s + 15·31-s + 110·33-s + 22·35-s + 5·37-s + 24·41-s + 40·43-s + 110·45-s − 9·47-s + 28·49-s + 130·51-s + 6·53-s + 22·55-s + 180·57-s + 28·59-s + 39·61-s + ⋯ |
| L(s) = 1 | + 5.77·3-s + 0.894·5-s + 4.15·7-s + 55/3·9-s + 3.31·11-s + 5.16·15-s + 3.15·17-s + 4.12·19-s + 24.0·21-s − 3.59·25-s + 42.3·27-s + 0.928·29-s + 2.69·31-s + 19.1·33-s + 3.71·35-s + 0.821·37-s + 3.74·41-s + 6.09·43-s + 16.3·45-s − 1.31·47-s + 4·49-s + 18.2·51-s + 0.824·53-s + 2.96·55-s + 23.8·57-s + 3.64·59-s + 4.99·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 3^{10} \cdot 23^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11979\times 10^{17}\) |
| Root analytic conductor: | \(7.11962\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 3^{10} \cdot 23^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(32496.95849\) |
| \(L(\frac12)\) | \(\approx\) | \(32496.95849\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{10} \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + 22 T^{2} - 38 T^{3} + 2 p^{3} T^{4} - 437 T^{5} + 2041 T^{6} - 3597 T^{7} + 13117 T^{8} - 22528 T^{9} + 70743 T^{10} - 22528 p T^{11} + 13117 p^{2} T^{12} - 3597 p^{3} T^{13} + 2041 p^{4} T^{14} - 437 p^{5} T^{15} + 2 p^{9} T^{16} - 38 p^{7} T^{17} + 22 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 11 T + 93 T^{2} - 572 T^{3} + 62 p^{2} T^{4} - 1947 p T^{5} + 55099 T^{6} - 197780 T^{7} + 652254 T^{8} - 1949904 T^{9} + 5402973 T^{10} - 1949904 p T^{11} + 652254 p^{2} T^{12} - 197780 p^{3} T^{13} + 55099 p^{4} T^{14} - 1947 p^{6} T^{15} + 62 p^{8} T^{16} - 572 p^{7} T^{17} + 93 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - p T + 126 T^{2} - 82 p T^{3} + 6258 T^{4} - 3110 p T^{5} + 179145 T^{6} - 72309 p T^{7} + 3375421 T^{8} - 1135553 p T^{9} + 44191489 T^{10} - 1135553 p^{2} T^{11} + 3375421 p^{2} T^{12} - 72309 p^{4} T^{13} + 179145 p^{4} T^{14} - 3110 p^{6} T^{15} + 6258 p^{6} T^{16} - 82 p^{8} T^{17} + 126 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 47 T^{2} + 88 T^{3} + 1128 T^{4} + 4246 T^{5} + 22253 T^{6} + 93984 T^{7} + 427871 T^{8} + 1389850 T^{9} + 6590093 T^{10} + 1389850 p T^{11} + 427871 p^{2} T^{12} + 93984 p^{3} T^{13} + 22253 p^{4} T^{14} + 4246 p^{5} T^{15} + 1128 p^{6} T^{16} + 88 p^{7} T^{17} + 47 p^{8} T^{18} + p^{10} T^{20} \) | |
| 17 | \( 1 - 13 T + 172 T^{2} - 1523 T^{3} + 12415 T^{4} - 84570 T^{5} + 527622 T^{6} - 2927125 T^{7} + 14993264 T^{8} - 69694564 T^{9} + 300917203 T^{10} - 69694564 p T^{11} + 14993264 p^{2} T^{12} - 2927125 p^{3} T^{13} + 527622 p^{4} T^{14} - 84570 p^{5} T^{15} + 12415 p^{6} T^{16} - 1523 p^{7} T^{17} + 172 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 18 T + 237 T^{2} - 2191 T^{3} + 918 p T^{4} - 116533 T^{5} + 715461 T^{6} - 3939292 T^{7} + 20529428 T^{8} - 98112585 T^{9} + 446027461 T^{10} - 98112585 p T^{11} + 20529428 p^{2} T^{12} - 3939292 p^{3} T^{13} + 715461 p^{4} T^{14} - 116533 p^{5} T^{15} + 918 p^{7} T^{16} - 2191 p^{7} T^{17} + 237 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 5 T + 209 T^{2} - 863 T^{3} + 20390 T^{4} - 71919 T^{5} + 1262944 T^{6} - 3915924 T^{7} + 56039682 T^{8} - 153774410 T^{9} + 1865992315 T^{10} - 153774410 p T^{11} + 56039682 p^{2} T^{12} - 3915924 p^{3} T^{13} + 1262944 p^{4} T^{14} - 71919 p^{5} T^{15} + 20390 p^{6} T^{16} - 863 p^{7} T^{17} + 209 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 15 T + 308 T^{2} - 3263 T^{3} + 39464 T^{4} - 328265 T^{5} + 2964213 T^{6} - 20394275 T^{7} + 149767446 T^{8} - 874945500 T^{9} + 5428040719 T^{10} - 874945500 p T^{11} + 149767446 p^{2} T^{12} - 20394275 p^{3} T^{13} + 2964213 p^{4} T^{14} - 328265 p^{5} T^{15} + 39464 p^{6} T^{16} - 3263 p^{7} T^{17} + 308 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 5 T + 102 T^{2} - 695 T^{3} + 9838 T^{4} - 57661 T^{5} + 601927 T^{6} - 3477327 T^{7} + 31911162 T^{8} - 162808890 T^{9} + 1262243889 T^{10} - 162808890 p T^{11} + 31911162 p^{2} T^{12} - 3477327 p^{3} T^{13} + 601927 p^{4} T^{14} - 57661 p^{5} T^{15} + 9838 p^{6} T^{16} - 695 p^{7} T^{17} + 102 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 24 T + 469 T^{2} - 6378 T^{3} + 77952 T^{4} - 796362 T^{5} + 7558724 T^{6} - 63372970 T^{7} + 500170554 T^{8} - 3560074826 T^{9} + 23947505649 T^{10} - 3560074826 p T^{11} + 500170554 p^{2} T^{12} - 63372970 p^{3} T^{13} + 7558724 p^{4} T^{14} - 796362 p^{5} T^{15} + 77952 p^{6} T^{16} - 6378 p^{7} T^{17} + 469 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 40 T + 1008 T^{2} - 18341 T^{3} + 269493 T^{4} - 3319000 T^{5} + 35494255 T^{6} - 334844656 T^{7} + 2828272361 T^{8} - 21515938324 T^{9} + 148252576809 T^{10} - 21515938324 p T^{11} + 2828272361 p^{2} T^{12} - 334844656 p^{3} T^{13} + 35494255 p^{4} T^{14} - 3319000 p^{5} T^{15} + 269493 p^{6} T^{16} - 18341 p^{7} T^{17} + 1008 p^{8} T^{18} - 40 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 9 T + 345 T^{2} + 2142 T^{3} + 51291 T^{4} + 231170 T^{5} + 4774721 T^{6} + 16929214 T^{7} + 329734077 T^{8} + 995101632 T^{9} + 17642029117 T^{10} + 995101632 p T^{11} + 329734077 p^{2} T^{12} + 16929214 p^{3} T^{13} + 4774721 p^{4} T^{14} + 231170 p^{5} T^{15} + 51291 p^{6} T^{16} + 2142 p^{7} T^{17} + 345 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 6 T + 281 T^{2} - 1897 T^{3} + 44106 T^{4} - 299971 T^{5} + 4684003 T^{6} - 30775624 T^{7} + 368838268 T^{8} - 2232067285 T^{9} + 22207735335 T^{10} - 2232067285 p T^{11} + 368838268 p^{2} T^{12} - 30775624 p^{3} T^{13} + 4684003 p^{4} T^{14} - 299971 p^{5} T^{15} + 44106 p^{6} T^{16} - 1897 p^{7} T^{17} + 281 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 28 T + 619 T^{2} - 9194 T^{3} + 113414 T^{4} - 1068189 T^{5} + 8120405 T^{6} - 41911313 T^{7} + 92967377 T^{8} + 1014802879 T^{9} - 12548499321 T^{10} + 1014802879 p T^{11} + 92967377 p^{2} T^{12} - 41911313 p^{3} T^{13} + 8120405 p^{4} T^{14} - 1068189 p^{5} T^{15} + 113414 p^{6} T^{16} - 9194 p^{7} T^{17} + 619 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 39 T + 1079 T^{2} - 21611 T^{3} + 359638 T^{4} - 5040677 T^{5} + 62115512 T^{6} - 677923128 T^{7} + 6687920061 T^{8} - 59755179455 T^{9} + 488730348355 T^{10} - 59755179455 p T^{11} + 6687920061 p^{2} T^{12} - 677923128 p^{3} T^{13} + 62115512 p^{4} T^{14} - 5040677 p^{5} T^{15} + 359638 p^{6} T^{16} - 21611 p^{7} T^{17} + 1079 p^{8} T^{18} - 39 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 32 T + 944 T^{2} - 17497 T^{3} + 298638 T^{4} - 3935843 T^{5} + 48862510 T^{6} - 505164663 T^{7} + 5068998107 T^{8} - 44237421544 T^{9} + 384856534163 T^{10} - 44237421544 p T^{11} + 5068998107 p^{2} T^{12} - 505164663 p^{3} T^{13} + 48862510 p^{4} T^{14} - 3935843 p^{5} T^{15} + 298638 p^{6} T^{16} - 17497 p^{7} T^{17} + 944 p^{8} T^{18} - 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 33 T + 785 T^{2} + 13860 T^{3} + 211173 T^{4} + 2781262 T^{5} + 33328173 T^{6} + 362779230 T^{7} + 3675346143 T^{8} + 34353291412 T^{9} + 300762736747 T^{10} + 34353291412 p T^{11} + 3675346143 p^{2} T^{12} + 362779230 p^{3} T^{13} + 33328173 p^{4} T^{14} + 2781262 p^{5} T^{15} + 211173 p^{6} T^{16} + 13860 p^{7} T^{17} + 785 p^{8} T^{18} + 33 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 50 T + 1651 T^{2} + 39316 T^{3} + 766206 T^{4} + 12506541 T^{5} + 177834959 T^{6} + 2220751133 T^{7} + 24777368289 T^{8} + 247372516083 T^{9} + 2228435826001 T^{10} + 247372516083 p T^{11} + 24777368289 p^{2} T^{12} + 2220751133 p^{3} T^{13} + 177834959 p^{4} T^{14} + 12506541 p^{5} T^{15} + 766206 p^{6} T^{16} + 39316 p^{7} T^{17} + 1651 p^{8} T^{18} + 50 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 33 T + 1024 T^{2} - 21307 T^{3} + 398997 T^{4} - 6189150 T^{5} + 86737015 T^{6} - 1074297829 T^{7} + 12101600296 T^{8} - 123424746093 T^{9} + 1149201764773 T^{10} - 123424746093 p T^{11} + 12101600296 p^{2} T^{12} - 1074297829 p^{3} T^{13} + 86737015 p^{4} T^{14} - 6189150 p^{5} T^{15} + 398997 p^{6} T^{16} - 21307 p^{7} T^{17} + 1024 p^{8} T^{18} - 33 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 29 T + 994 T^{2} - 19075 T^{3} + 387681 T^{4} - 5691324 T^{5} + 85952923 T^{6} - 1023170783 T^{7} + 12393471730 T^{8} - 122631197781 T^{9} + 1229894410679 T^{10} - 122631197781 p T^{11} + 12393471730 p^{2} T^{12} - 1023170783 p^{3} T^{13} + 85952923 p^{4} T^{14} - 5691324 p^{5} T^{15} + 387681 p^{6} T^{16} - 19075 p^{7} T^{17} + 994 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 17 T + 599 T^{2} - 8233 T^{3} + 175174 T^{4} - 2063669 T^{5} + 33268406 T^{6} - 342041032 T^{7} + 4541311261 T^{8} - 40924169769 T^{9} + 464385612245 T^{10} - 40924169769 p T^{11} + 4541311261 p^{2} T^{12} - 342041032 p^{3} T^{13} + 33268406 p^{4} T^{14} - 2063669 p^{5} T^{15} + 175174 p^{6} T^{16} - 8233 p^{7} T^{17} + 599 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 46 T + 1613 T^{2} - 40084 T^{3} + 851448 T^{4} - 15101063 T^{5} + 238897757 T^{6} - 3321083869 T^{7} + 41965008981 T^{8} - 475354585259 T^{9} + 4931157443573 T^{10} - 475354585259 p T^{11} + 41965008981 p^{2} T^{12} - 3321083869 p^{3} T^{13} + 238897757 p^{4} T^{14} - 15101063 p^{5} T^{15} + 851448 p^{6} T^{16} - 40084 p^{7} T^{17} + 1613 p^{8} T^{18} - 46 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.73907515391134835789273662196, −2.62894505839427235890974994476, −2.60874754557376814545413435262, −2.57653543953400112490516875862, −2.51783334343525954354960929864, −2.04164343623286110099561845006, −2.02172570993907659245615502199, −2.01496636782259636477133426578, −2.00021874024667581054446591280, −1.93398507043432496766137069638, −1.91749140676195464094235053215, −1.78884814285047352506616815109, −1.78430168103193198238438334311, −1.66166249865008688746147765810, −1.61429094421111388759665647659, −1.15951864253438493896943040157, −1.12944784821482962784186559573, −1.03931986360942162608214510282, −0.996047744145417379682111311490, −0.991463123847267104916797406686, −0.918588765620667609138690879049, −0.891939153300220256191331250997, −0.68115511272940066502608310526, −0.66932091577241356965660580020, −0.65153632609238689528768348354, 0.65153632609238689528768348354, 0.66932091577241356965660580020, 0.68115511272940066502608310526, 0.891939153300220256191331250997, 0.918588765620667609138690879049, 0.991463123847267104916797406686, 0.996047744145417379682111311490, 1.03931986360942162608214510282, 1.12944784821482962784186559573, 1.15951864253438493896943040157, 1.61429094421111388759665647659, 1.66166249865008688746147765810, 1.78430168103193198238438334311, 1.78884814285047352506616815109, 1.91749140676195464094235053215, 1.93398507043432496766137069638, 2.00021874024667581054446591280, 2.01496636782259636477133426578, 2.02172570993907659245615502199, 2.04164343623286110099561845006, 2.51783334343525954354960929864, 2.57653543953400112490516875862, 2.60874754557376814545413435262, 2.62894505839427235890974994476, 2.73907515391134835789273662196
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.