Dirichlet series
| L(s) = 1 | + 11·3-s + 10·5-s + 7-s + 66·9-s + 11-s + 10·13-s + 110·15-s + 17·17-s − 2·19-s + 11·21-s + 3·23-s + 33·25-s + 286·27-s + 17·29-s + 15·31-s + 11·33-s + 10·35-s + 4·37-s + 110·39-s + 16·41-s − 10·43-s + 660·45-s + 16·47-s − 27·49-s + 187·51-s + 42·53-s + 10·55-s + ⋯ |
| L(s) = 1 | + 6.35·3-s + 4.47·5-s + 0.377·7-s + 22·9-s + 0.301·11-s + 2.77·13-s + 28.4·15-s + 4.12·17-s − 0.458·19-s + 2.40·21-s + 0.625·23-s + 33/5·25-s + 55.0·27-s + 3.15·29-s + 2.69·31-s + 1.91·33-s + 1.69·35-s + 0.657·37-s + 17.6·39-s + 2.49·41-s − 1.52·43-s + 98.3·45-s + 2.33·47-s − 3.85·49-s + 26.1·51-s + 5.76·53-s + 1.34·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{44} \cdot 3^{11} \cdot 167^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{44} \cdot 3^{11} \cdot 167^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{44} \cdot 3^{11} \cdot 167^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.38895\times 10^{19}\) |
| Root analytic conductor: | \(8.00050\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{44} \cdot 3^{11} \cdot 167^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(70701.33138\) |
| \(L(\frac12)\) | \(\approx\) | \(70701.33138\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{11} \) | |
| 167 | \( ( 1 + T )^{11} \) | |
| good | 5 | \( 1 - 2 p T + 67 T^{2} - 69 p T^{3} + 306 p T^{4} - 5914 T^{5} + 4123 p T^{6} - 65174 T^{7} + 189802 T^{8} - 509486 T^{9} + 254337 p T^{10} - 2945398 T^{11} + 254337 p^{2} T^{12} - 509486 p^{2} T^{13} + 189802 p^{3} T^{14} - 65174 p^{4} T^{15} + 4123 p^{6} T^{16} - 5914 p^{6} T^{17} + 306 p^{8} T^{18} - 69 p^{9} T^{19} + 67 p^{9} T^{20} - 2 p^{11} T^{21} + p^{11} T^{22} \) |
| 7 | \( 1 - T + 4 p T^{2} - 24 T^{3} + 408 T^{4} - 321 T^{5} + 4798 T^{6} - 3540 T^{7} + 6773 p T^{8} - 31342 T^{9} + 378850 T^{10} - 230184 T^{11} + 378850 p T^{12} - 31342 p^{2} T^{13} + 6773 p^{4} T^{14} - 3540 p^{4} T^{15} + 4798 p^{5} T^{16} - 321 p^{6} T^{17} + 408 p^{7} T^{18} - 24 p^{8} T^{19} + 4 p^{10} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 - T + 48 T^{2} - 26 T^{3} + 1293 T^{4} - 412 T^{5} + 25538 T^{6} - 4366 T^{7} + 402125 T^{8} - 33083 T^{9} + 5243181 T^{10} - 242368 T^{11} + 5243181 p T^{12} - 33083 p^{2} T^{13} + 402125 p^{3} T^{14} - 4366 p^{4} T^{15} + 25538 p^{5} T^{16} - 412 p^{6} T^{17} + 1293 p^{7} T^{18} - 26 p^{8} T^{19} + 48 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 - 10 T + 116 T^{2} - 816 T^{3} + 5728 T^{4} - 31413 T^{5} + 169547 T^{6} - 774510 T^{7} + 3520316 T^{8} - 14058633 T^{9} + 56398720 T^{10} - 202572420 T^{11} + 56398720 p T^{12} - 14058633 p^{2} T^{13} + 3520316 p^{3} T^{14} - 774510 p^{4} T^{15} + 169547 p^{5} T^{16} - 31413 p^{6} T^{17} + 5728 p^{7} T^{18} - 816 p^{8} T^{19} + 116 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 - p T + 241 T^{2} - 2426 T^{3} + 21464 T^{4} - 159918 T^{5} + 63558 p T^{6} - 6482792 T^{7} + 35861503 T^{8} - 180071453 T^{9} + 840324305 T^{10} - 211161236 p T^{11} + 840324305 p T^{12} - 180071453 p^{2} T^{13} + 35861503 p^{3} T^{14} - 6482792 p^{4} T^{15} + 63558 p^{6} T^{16} - 159918 p^{6} T^{17} + 21464 p^{7} T^{18} - 2426 p^{8} T^{19} + 241 p^{9} T^{20} - p^{11} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 + 2 T + 104 T^{2} + 96 T^{3} + 5444 T^{4} - 761 T^{5} + 186103 T^{6} - 224246 T^{7} + 4770170 T^{8} - 9884721 T^{9} + 101541836 T^{10} - 240128532 T^{11} + 101541836 p T^{12} - 9884721 p^{2} T^{13} + 4770170 p^{3} T^{14} - 224246 p^{4} T^{15} + 186103 p^{5} T^{16} - 761 p^{6} T^{17} + 5444 p^{7} T^{18} + 96 p^{8} T^{19} + 104 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 3 T + 162 T^{2} - 512 T^{3} + 12407 T^{4} - 43808 T^{5} + 605456 T^{6} - 2403850 T^{7} + 21488859 T^{8} - 91406541 T^{9} + 600918697 T^{10} - 2479545100 T^{11} + 600918697 p T^{12} - 91406541 p^{2} T^{13} + 21488859 p^{3} T^{14} - 2403850 p^{4} T^{15} + 605456 p^{5} T^{16} - 43808 p^{6} T^{17} + 12407 p^{7} T^{18} - 512 p^{8} T^{19} + 162 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 - 17 T + 374 T^{2} - 4462 T^{3} + 57451 T^{4} - 534092 T^{5} + 5082040 T^{6} - 38682606 T^{7} + 295198523 T^{8} - 1885119491 T^{9} + 11965413375 T^{10} - 64710360584 T^{11} + 11965413375 p T^{12} - 1885119491 p^{2} T^{13} + 295198523 p^{3} T^{14} - 38682606 p^{4} T^{15} + 5082040 p^{5} T^{16} - 534092 p^{6} T^{17} + 57451 p^{7} T^{18} - 4462 p^{8} T^{19} + 374 p^{9} T^{20} - 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 15 T + 277 T^{2} - 3289 T^{3} + 37064 T^{4} - 354644 T^{5} + 3091453 T^{6} - 24533802 T^{7} + 178218002 T^{8} - 1198477245 T^{9} + 7456864249 T^{10} - 43067489466 T^{11} + 7456864249 p T^{12} - 1198477245 p^{2} T^{13} + 178218002 p^{3} T^{14} - 24533802 p^{4} T^{15} + 3091453 p^{5} T^{16} - 354644 p^{6} T^{17} + 37064 p^{7} T^{18} - 3289 p^{8} T^{19} + 277 p^{9} T^{20} - 15 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - 4 T + 183 T^{2} - 607 T^{3} + 15052 T^{4} - 47184 T^{5} + 792027 T^{6} - 3114852 T^{7} + 34303738 T^{8} - 186828452 T^{9} + 1400308163 T^{10} - 8356445354 T^{11} + 1400308163 p T^{12} - 186828452 p^{2} T^{13} + 34303738 p^{3} T^{14} - 3114852 p^{4} T^{15} + 792027 p^{5} T^{16} - 47184 p^{6} T^{17} + 15052 p^{7} T^{18} - 607 p^{8} T^{19} + 183 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 - 16 T + 399 T^{2} - 4853 T^{3} + 70135 T^{4} - 690939 T^{5} + 7427693 T^{6} - 61625762 T^{7} + 540949428 T^{8} - 3875396693 T^{9} + 29071797776 T^{10} - 182118029858 T^{11} + 29071797776 p T^{12} - 3875396693 p^{2} T^{13} + 540949428 p^{3} T^{14} - 61625762 p^{4} T^{15} + 7427693 p^{5} T^{16} - 690939 p^{6} T^{17} + 70135 p^{7} T^{18} - 4853 p^{8} T^{19} + 399 p^{9} T^{20} - 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 + 10 T + 279 T^{2} + 2075 T^{3} + 36023 T^{4} + 221003 T^{5} + 3082429 T^{6} + 16403326 T^{7} + 200110078 T^{8} + 952121859 T^{9} + 10478092890 T^{10} + 45165864910 T^{11} + 10478092890 p T^{12} + 952121859 p^{2} T^{13} + 200110078 p^{3} T^{14} + 16403326 p^{4} T^{15} + 3082429 p^{5} T^{16} + 221003 p^{6} T^{17} + 36023 p^{7} T^{18} + 2075 p^{8} T^{19} + 279 p^{9} T^{20} + 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 - 16 T + 307 T^{2} - 3330 T^{3} + 810 p T^{4} - 298909 T^{5} + 2385155 T^{6} - 12604810 T^{7} + 66749748 T^{8} - 88508431 T^{9} - 248658699 T^{10} + 10612004112 T^{11} - 248658699 p T^{12} - 88508431 p^{2} T^{13} + 66749748 p^{3} T^{14} - 12604810 p^{4} T^{15} + 2385155 p^{5} T^{16} - 298909 p^{6} T^{17} + 810 p^{8} T^{18} - 3330 p^{8} T^{19} + 307 p^{9} T^{20} - 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 - 42 T + 1173 T^{2} - 23620 T^{3} + 389854 T^{4} - 5399067 T^{5} + 65490503 T^{6} - 703633882 T^{7} + 6846296048 T^{8} - 60676194697 T^{9} + 496287074001 T^{10} - 3748573238368 T^{11} + 496287074001 p T^{12} - 60676194697 p^{2} T^{13} + 6846296048 p^{3} T^{14} - 703633882 p^{4} T^{15} + 65490503 p^{5} T^{16} - 5399067 p^{6} T^{17} + 389854 p^{7} T^{18} - 23620 p^{8} T^{19} + 1173 p^{9} T^{20} - 42 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 - 2 T + 505 T^{2} - 666 T^{3} + 120674 T^{4} - 94581 T^{5} + 18110815 T^{6} - 7300032 T^{7} + 1907206020 T^{8} - 344138969 T^{9} + 148634314931 T^{10} - 15352893564 T^{11} + 148634314931 p T^{12} - 344138969 p^{2} T^{13} + 1907206020 p^{3} T^{14} - 7300032 p^{4} T^{15} + 18110815 p^{5} T^{16} - 94581 p^{6} T^{17} + 120674 p^{7} T^{18} - 666 p^{8} T^{19} + 505 p^{9} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 - 12 T + 340 T^{2} - 3384 T^{3} + 57488 T^{4} - 463101 T^{5} + 6208719 T^{6} - 41463550 T^{7} + 488568268 T^{8} - 2821350367 T^{9} + 32150979308 T^{10} - 172424859412 T^{11} + 32150979308 p T^{12} - 2821350367 p^{2} T^{13} + 488568268 p^{3} T^{14} - 41463550 p^{4} T^{15} + 6208719 p^{5} T^{16} - 463101 p^{6} T^{17} + 57488 p^{7} T^{18} - 3384 p^{8} T^{19} + 340 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 - T + 436 T^{2} - 865 T^{3} + 97490 T^{4} - 247038 T^{5} + 14590774 T^{6} - 40768042 T^{7} + 1610450641 T^{8} - 4515496635 T^{9} + 137204831826 T^{10} - 355748041550 T^{11} + 137204831826 p T^{12} - 4515496635 p^{2} T^{13} + 1610450641 p^{3} T^{14} - 40768042 p^{4} T^{15} + 14590774 p^{5} T^{16} - 247038 p^{6} T^{17} + 97490 p^{7} T^{18} - 865 p^{8} T^{19} + 436 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 - 9 T + 407 T^{2} - 2019 T^{3} + 72682 T^{4} - 216594 T^{5} + 9495944 T^{6} - 23315926 T^{7} + 1016193411 T^{8} - 2055559861 T^{9} + 85180979199 T^{10} - 141342461934 T^{11} + 85180979199 p T^{12} - 2055559861 p^{2} T^{13} + 1016193411 p^{3} T^{14} - 23315926 p^{4} T^{15} + 9495944 p^{5} T^{16} - 216594 p^{6} T^{17} + 72682 p^{7} T^{18} - 2019 p^{8} T^{19} + 407 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 - 24 T + 832 T^{2} - 14672 T^{3} + 296314 T^{4} - 4148141 T^{5} + 62082611 T^{6} - 719899178 T^{7} + 8699232388 T^{8} - 85544428603 T^{9} + 867998838438 T^{10} - 7310385685164 T^{11} + 867998838438 p T^{12} - 85544428603 p^{2} T^{13} + 8699232388 p^{3} T^{14} - 719899178 p^{4} T^{15} + 62082611 p^{5} T^{16} - 4148141 p^{6} T^{17} + 296314 p^{7} T^{18} - 14672 p^{8} T^{19} + 832 p^{9} T^{20} - 24 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 30 T + 645 T^{2} - 9619 T^{3} + 123851 T^{4} - 1379707 T^{5} + 14867779 T^{6} - 154181526 T^{7} + 19888740 p T^{8} - 15355190883 T^{9} + 144073470752 T^{10} - 1303460329558 T^{11} + 144073470752 p T^{12} - 15355190883 p^{2} T^{13} + 19888740 p^{4} T^{14} - 154181526 p^{4} T^{15} + 14867779 p^{5} T^{16} - 1379707 p^{6} T^{17} + 123851 p^{7} T^{18} - 9619 p^{8} T^{19} + 645 p^{9} T^{20} - 30 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 + 16 T + 559 T^{2} + 8890 T^{3} + 166834 T^{4} + 2332409 T^{5} + 33323265 T^{6} + 397711016 T^{7} + 4765849894 T^{8} + 49629010023 T^{9} + 511273556061 T^{10} + 4708369901132 T^{11} + 511273556061 p T^{12} + 49629010023 p^{2} T^{13} + 4765849894 p^{3} T^{14} + 397711016 p^{4} T^{15} + 33323265 p^{5} T^{16} + 2332409 p^{6} T^{17} + 166834 p^{7} T^{18} + 8890 p^{8} T^{19} + 559 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 - 37 T + 1176 T^{2} - 25175 T^{3} + 484854 T^{4} - 7662396 T^{5} + 112691726 T^{6} - 1458814172 T^{7} + 17904001283 T^{8} - 198977030567 T^{9} + 2106943642192 T^{10} - 20364870673674 T^{11} + 2106943642192 p T^{12} - 198977030567 p^{2} T^{13} + 17904001283 p^{3} T^{14} - 1458814172 p^{4} T^{15} + 112691726 p^{5} T^{16} - 7662396 p^{6} T^{17} + 484854 p^{7} T^{18} - 25175 p^{8} T^{19} + 1176 p^{9} T^{20} - 37 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 - 4 T + 512 T^{2} - 1207 T^{3} + 132392 T^{4} - 161794 T^{5} + 24030516 T^{6} - 13362994 T^{7} + 3429379285 T^{8} - 700457026 T^{9} + 400079268110 T^{10} - 34763160478 T^{11} + 400079268110 p T^{12} - 700457026 p^{2} T^{13} + 3429379285 p^{3} T^{14} - 13362994 p^{4} T^{15} + 24030516 p^{5} T^{16} - 161794 p^{6} T^{17} + 132392 p^{7} T^{18} - 1207 p^{8} T^{19} + 512 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.54484615811875143001192751733, −2.43301935993732376572678658112, −2.35873449901541051266872847303, −2.17372481983245762869181118046, −2.05208593552907738799192087809, −1.95837277679158386426423025315, −1.93114431428867355485984028399, −1.87479976127938756280110250011, −1.82076957964637101252894348918, −1.79033849915491980871183302615, −1.75010491873923793225083766154, −1.73478722833450916906733578568, −1.71913000739652613781359198547, −1.58164215429320909551707958839, −1.15224419493487414603613706891, −1.14934157163763247527317651852, −1.04359694346516040990901997101, −0.996345833357379752189259597592, −0.964009810244240271423443631255, −0.902578889882153773795741859947, −0.885872249558148497873950069949, −0.77070989295029803958593360929, −0.67631928690572455228011411187, −0.53048879360008684140098530121, −0.44973376264797522626689594614, 0.44973376264797522626689594614, 0.53048879360008684140098530121, 0.67631928690572455228011411187, 0.77070989295029803958593360929, 0.885872249558148497873950069949, 0.902578889882153773795741859947, 0.964009810244240271423443631255, 0.996345833357379752189259597592, 1.04359694346516040990901997101, 1.14934157163763247527317651852, 1.15224419493487414603613706891, 1.58164215429320909551707958839, 1.71913000739652613781359198547, 1.73478722833450916906733578568, 1.75010491873923793225083766154, 1.79033849915491980871183302615, 1.82076957964637101252894348918, 1.87479976127938756280110250011, 1.93114431428867355485984028399, 1.95837277679158386426423025315, 2.05208593552907738799192087809, 2.17372481983245762869181118046, 2.35873449901541051266872847303, 2.43301935993732376572678658112, 2.54484615811875143001192751733
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.