Dirichlet series
| L(s) = 1 | − 6·4-s − 45·9-s + 110·11-s + 151·16-s + 674·19-s + 306·29-s + 526·31-s + 270·36-s − 176·41-s − 660·44-s − 666·49-s − 820·59-s − 2.26e3·61-s − 200·64-s + 2.49e3·71-s − 4.04e3·76-s − 4.51e3·79-s + 1.21e3·81-s − 694·89-s − 4.95e3·99-s − 810·101-s − 1.82e3·109-s − 1.83e3·116-s + 6.65e3·121-s − 3.15e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/4·4-s − 5/3·9-s + 3.01·11-s + 2.35·16-s + 8.13·19-s + 1.95·29-s + 3.04·31-s + 5/4·36-s − 0.670·41-s − 2.26·44-s − 1.94·49-s − 1.80·59-s − 4.74·61-s − 0.390·64-s + 4.17·71-s − 6.10·76-s − 6.43·79-s + 5/3·81-s − 0.826·89-s − 5.02·99-s − 0.798·101-s − 1.60·109-s − 1.46·116-s + 5·121-s − 2.28·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{20} \cdot 11^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.46792\times 10^{16}\) |
| Root analytic conductor: | \(6.97686\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{20} \cdot 11^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(26.64104025\) |
| \(L(\frac12)\) | \(\approx\) | \(26.64104025\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{2} T^{2} )^{5} \) |
| 5 | \( 1 \) | |
| 11 | \( ( 1 - p T )^{10} \) | |
| good | 2 | \( 1 + 3 p T^{2} - 115 T^{4} - 349 p^{2} T^{6} + 1009 p^{2} T^{8} + 235 p^{9} T^{10} + 1009 p^{8} T^{12} - 349 p^{14} T^{14} - 115 p^{18} T^{16} + 3 p^{25} T^{18} + p^{30} T^{20} \) |
| 7 | \( 1 + 666 T^{2} + 37645 p T^{4} + 71564784 T^{6} + 874433789 p^{2} T^{8} + 7258258130 p^{4} T^{10} + 874433789 p^{8} T^{12} + 71564784 p^{12} T^{14} + 37645 p^{19} T^{16} + 666 p^{24} T^{18} + p^{30} T^{20} \) | |
| 13 | \( 1 - 8799 T^{2} + 45948304 T^{4} - 171681976941 T^{6} + 518179386816571 T^{8} - 1258379477992000880 T^{10} + 518179386816571 p^{6} T^{12} - 171681976941 p^{12} T^{14} + 45948304 p^{18} T^{16} - 8799 p^{24} T^{18} + p^{30} T^{20} \) | |
| 17 | \( 1 - 24920 T^{2} + 288998262 T^{4} - 2080177839230 T^{6} + 11067911766515145 T^{8} - 53503539758089881940 T^{10} + 11067911766515145 p^{6} T^{12} - 2080177839230 p^{12} T^{14} + 288998262 p^{18} T^{16} - 24920 p^{24} T^{18} + p^{30} T^{20} \) | |
| 19 | \( ( 1 - 337 T + 60817 T^{2} - 7592846 T^{3} + 750397753 T^{4} - 64855829999 T^{5} + 750397753 p^{3} T^{6} - 7592846 p^{6} T^{7} + 60817 p^{9} T^{8} - 337 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 23 | \( 1 - 79355 T^{2} + 2928380128 T^{4} - 67811394206105 T^{6} + 1138216271752329771 T^{8} - \)\(15\!\cdots\!80\)\( T^{10} + 1138216271752329771 p^{6} T^{12} - 67811394206105 p^{12} T^{14} + 2928380128 p^{18} T^{16} - 79355 p^{24} T^{18} + p^{30} T^{20} \) | |
| 29 | \( ( 1 - 153 T + 88541 T^{2} - 13903604 T^{3} + 3631769750 T^{4} - 497564300566 T^{5} + 3631769750 p^{3} T^{6} - 13903604 p^{6} T^{7} + 88541 p^{9} T^{8} - 153 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 31 | \( ( 1 - 263 T + 148596 T^{2} - 27848973 T^{3} + 8750489917 T^{4} - 1199133615296 T^{5} + 8750489917 p^{3} T^{6} - 27848973 p^{6} T^{7} + 148596 p^{9} T^{8} - 263 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 37 | \( 1 - 336556 T^{2} + 51765473206 T^{4} - 4923644149697906 T^{6} + \)\(33\!\cdots\!57\)\( T^{8} - \)\(18\!\cdots\!36\)\( T^{10} + \)\(33\!\cdots\!57\)\( p^{6} T^{12} - 4923644149697906 p^{12} T^{14} + 51765473206 p^{18} T^{16} - 336556 p^{24} T^{18} + p^{30} T^{20} \) | |
| 41 | \( ( 1 + 88 T + 146826 T^{2} + 11079990 T^{3} + 14142005749 T^{4} + 376619177092 T^{5} + 14142005749 p^{3} T^{6} + 11079990 p^{6} T^{7} + 146826 p^{9} T^{8} + 88 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 43 | \( 1 - 586019 T^{2} + 157534290416 T^{4} - 26135055894811761 T^{6} + \)\(30\!\cdots\!99\)\( T^{8} - \)\(27\!\cdots\!60\)\( T^{10} + \)\(30\!\cdots\!99\)\( p^{6} T^{12} - 26135055894811761 p^{12} T^{14} + 157534290416 p^{18} T^{16} - 586019 p^{24} T^{18} + p^{30} T^{20} \) | |
| 47 | \( 1 - 591524 T^{2} + 183953984862 T^{4} - 38453280293834010 T^{6} + \)\(59\!\cdots\!77\)\( T^{8} - \)\(69\!\cdots\!52\)\( T^{10} + \)\(59\!\cdots\!77\)\( p^{6} T^{12} - 38453280293834010 p^{12} T^{14} + 183953984862 p^{18} T^{16} - 591524 p^{24} T^{18} + p^{30} T^{20} \) | |
| 53 | \( 1 - 671846 T^{2} + 227325611845 T^{4} - 52544028582802472 T^{6} + \)\(96\!\cdots\!26\)\( T^{8} - \)\(15\!\cdots\!64\)\( T^{10} + \)\(96\!\cdots\!26\)\( p^{6} T^{12} - 52544028582802472 p^{12} T^{14} + 227325611845 p^{18} T^{16} - 671846 p^{24} T^{18} + p^{30} T^{20} \) | |
| 59 | \( ( 1 + 410 T + 462506 T^{2} + 220493866 T^{3} + 92237734701 T^{4} + 56931666042088 T^{5} + 92237734701 p^{3} T^{6} + 220493866 p^{6} T^{7} + 462506 p^{9} T^{8} + 410 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 61 | \( ( 1 + 1130 T + 660198 T^{2} + 221773540 T^{3} + 106549290025 T^{4} + 52015880337732 T^{5} + 106549290025 p^{3} T^{6} + 221773540 p^{6} T^{7} + 660198 p^{9} T^{8} + 1130 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 67 | \( 1 - 1579924 T^{2} + 1262992599350 T^{4} - 684012874001661786 T^{6} + \)\(28\!\cdots\!21\)\( T^{8} - \)\(93\!\cdots\!60\)\( T^{10} + \)\(28\!\cdots\!21\)\( p^{6} T^{12} - 684012874001661786 p^{12} T^{14} + 1262992599350 p^{18} T^{16} - 1579924 p^{24} T^{18} + p^{30} T^{20} \) | |
| 71 | \( ( 1 - 1249 T + 889878 T^{2} - 562304447 T^{3} + 478254611297 T^{4} - 328879704249920 T^{5} + 478254611297 p^{3} T^{6} - 562304447 p^{6} T^{7} + 889878 p^{9} T^{8} - 1249 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 73 | \( 1 - 2396570 T^{2} + 2775222636093 T^{4} - 2089065123571938040 T^{6} + \)\(11\!\cdots\!26\)\( T^{8} - \)\(50\!\cdots\!80\)\( T^{10} + \)\(11\!\cdots\!26\)\( p^{6} T^{12} - 2089065123571938040 p^{12} T^{14} + 2775222636093 p^{18} T^{16} - 2396570 p^{24} T^{18} + p^{30} T^{20} \) | |
| 79 | \( ( 1 + 2258 T + 3808964 T^{2} + 4519274572 T^{3} + 55752550289 p T^{4} + 3357879659061380 T^{5} + 55752550289 p^{4} T^{6} + 4519274572 p^{6} T^{7} + 3808964 p^{9} T^{8} + 2258 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 83 | \( 1 - 2734409 T^{2} + 3420948324585 T^{4} - 2500137100228655876 T^{6} + \)\(12\!\cdots\!34\)\( T^{8} - \)\(60\!\cdots\!90\)\( T^{10} + \)\(12\!\cdots\!34\)\( p^{6} T^{12} - 2500137100228655876 p^{12} T^{14} + 3420948324585 p^{18} T^{16} - 2734409 p^{24} T^{18} + p^{30} T^{20} \) | |
| 89 | \( ( 1 + 347 T + 2521371 T^{2} + 1045079328 T^{3} + 3039141425436 T^{4} + 1087844980301290 T^{5} + 3039141425436 p^{3} T^{6} + 1045079328 p^{6} T^{7} + 2521371 p^{9} T^{8} + 347 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 97 | \( 1 - 6199965 T^{2} + 18565818255919 T^{4} - 35631592704346198158 T^{6} + \)\(49\!\cdots\!17\)\( T^{8} - \)\(51\!\cdots\!59\)\( T^{10} + \)\(49\!\cdots\!17\)\( p^{6} T^{12} - 35631592704346198158 p^{12} T^{14} + 18565818255919 p^{18} T^{16} - 6199965 p^{24} T^{18} + p^{30} T^{20} \) | |
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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
−3.38648203844714888003968245524, −3.05996392780641534013783146702, −2.94173697628743702137723110906, −2.92321873877968889744229476404, −2.90966280773409080645066010986, −2.86004066191563273688639479159, −2.81219261199426507296403318809, −2.68739206990591868812017076224, −2.64163295272357227902005371340, −2.42058106148993245896441180675, −1.84555120540693293487384448581, −1.78347524130547476582457536545, −1.60861084690180098831285919510, −1.55963919262970270170784017725, −1.53879452056113438302429930086, −1.53451628920443254621639679251, −1.28405718233733019301666082320, −1.09015359937711214987059310070, −0.999635919620517361532797084545, −0.987852659991446679850182491038, −0.67548368586267851988246796038, −0.63698376843907699756728789852, −0.52265031199242295407682000556, −0.50139411706385640992241903535, −0.11877992852208674548357863349, 0.11877992852208674548357863349, 0.50139411706385640992241903535, 0.52265031199242295407682000556, 0.63698376843907699756728789852, 0.67548368586267851988246796038, 0.987852659991446679850182491038, 0.999635919620517361532797084545, 1.09015359937711214987059310070, 1.28405718233733019301666082320, 1.53451628920443254621639679251, 1.53879452056113438302429930086, 1.55963919262970270170784017725, 1.60861084690180098831285919510, 1.78347524130547476582457536545, 1.84555120540693293487384448581, 2.42058106148993245896441180675, 2.64163295272357227902005371340, 2.68739206990591868812017076224, 2.81219261199426507296403318809, 2.86004066191563273688639479159, 2.90966280773409080645066010986, 2.92321873877968889744229476404, 2.94173697628743702137723110906, 3.05996392780641534013783146702, 3.38648203844714888003968245524
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.