Dirichlet series
| L(s) = 1 | + 11·2-s + 3·3-s + 66·4-s + 11·5-s + 33·6-s + 5·7-s + 286·8-s + 9-s + 121·10-s + 11·11-s + 198·12-s + 13-s + 55·14-s + 33·15-s + 1.00e3·16-s + 8·17-s + 11·18-s + 19-s + 726·20-s + 15·21-s + 121·22-s + 20·23-s + 858·24-s + 66·25-s + 11·26-s − 3·27-s + 330·28-s + ⋯ |
| L(s) = 1 | + 7.77·2-s + 1.73·3-s + 33·4-s + 4.91·5-s + 13.4·6-s + 1.88·7-s + 101.·8-s + 1/3·9-s + 38.2·10-s + 3.31·11-s + 57.1·12-s + 0.277·13-s + 14.6·14-s + 8.52·15-s + 250.·16-s + 1.94·17-s + 2.59·18-s + 0.229·19-s + 162.·20-s + 3.27·21-s + 25.7·22-s + 4.17·23-s + 175.·24-s + 66/5·25-s + 2.15·26-s − 0.577·27-s + 62.3·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.23110\times 10^{17}\) |
| Root analytic conductor: | \(6.14566\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.248241860\times10^{6}\) |
| \(L(\frac12)\) | \(\approx\) | \(2.248241860\times10^{6}\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{11} \) |
| 5 | \( ( 1 - T )^{11} \) | |
| 11 | \( ( 1 - T )^{11} \) | |
| 43 | \( ( 1 + T )^{11} \) | |
| good | 3 | \( 1 - p T + 8 T^{2} - 2 p^{2} T^{3} + 4 p^{2} T^{4} - 59 T^{5} + 124 T^{6} - 23 p^{2} T^{7} + 38 p^{2} T^{8} - 662 T^{9} + 413 p T^{10} - 1910 T^{11} + 413 p^{2} T^{12} - 662 p^{2} T^{13} + 38 p^{5} T^{14} - 23 p^{6} T^{15} + 124 p^{5} T^{16} - 59 p^{6} T^{17} + 4 p^{9} T^{18} - 2 p^{10} T^{19} + 8 p^{9} T^{20} - p^{11} T^{21} + p^{11} T^{22} \) |
| 7 | \( 1 - 5 T + 50 T^{2} - 190 T^{3} + 1196 T^{4} - 3839 T^{5} + 18826 T^{6} - 52789 T^{7} + 217656 T^{8} - 1578 p^{3} T^{9} + 1940685 T^{10} - 4289798 T^{11} + 1940685 p T^{12} - 1578 p^{5} T^{13} + 217656 p^{3} T^{14} - 52789 p^{4} T^{15} + 18826 p^{5} T^{16} - 3839 p^{6} T^{17} + 1196 p^{7} T^{18} - 190 p^{8} T^{19} + 50 p^{9} T^{20} - 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 - T + 81 T^{2} - 6 p T^{3} + 3405 T^{4} - 241 p T^{5} + 96573 T^{6} - 84232 T^{7} + 2036230 T^{8} - 1653738 T^{9} + 33403870 T^{10} - 24588436 T^{11} + 33403870 p T^{12} - 1653738 p^{2} T^{13} + 2036230 p^{3} T^{14} - 84232 p^{4} T^{15} + 96573 p^{5} T^{16} - 241 p^{7} T^{17} + 3405 p^{7} T^{18} - 6 p^{9} T^{19} + 81 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 - 8 T + 6 p T^{2} - 673 T^{3} + 5299 T^{4} - 29681 T^{5} + 185052 T^{6} - 912175 T^{7} + 4851713 T^{8} - 21430231 T^{9} + 101099041 T^{10} - 403343552 T^{11} + 101099041 p T^{12} - 21430231 p^{2} T^{13} + 4851713 p^{3} T^{14} - 912175 p^{4} T^{15} + 185052 p^{5} T^{16} - 29681 p^{6} T^{17} + 5299 p^{7} T^{18} - 673 p^{8} T^{19} + 6 p^{10} T^{20} - 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 - T + 102 T^{2} - 46 T^{3} + 5342 T^{4} - 1261 T^{5} + 194504 T^{6} - 48101 T^{7} + 5474534 T^{8} - 1617136 T^{9} + 124760351 T^{10} - 37040174 T^{11} + 124760351 p T^{12} - 1617136 p^{2} T^{13} + 5474534 p^{3} T^{14} - 48101 p^{4} T^{15} + 194504 p^{5} T^{16} - 1261 p^{6} T^{17} + 5342 p^{7} T^{18} - 46 p^{8} T^{19} + 102 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 20 T + 325 T^{2} - 3740 T^{3} + 37775 T^{4} - 320356 T^{5} + 2467139 T^{6} - 16860976 T^{7} + 4638182 p T^{8} - 613971656 T^{9} + 3305956354 T^{10} - 16363048296 T^{11} + 3305956354 p T^{12} - 613971656 p^{2} T^{13} + 4638182 p^{4} T^{14} - 16860976 p^{4} T^{15} + 2467139 p^{5} T^{16} - 320356 p^{6} T^{17} + 37775 p^{7} T^{18} - 3740 p^{8} T^{19} + 325 p^{9} T^{20} - 20 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 - 9 T + 162 T^{2} - 1053 T^{3} + 12166 T^{4} - 63513 T^{5} + 608249 T^{6} - 2695188 T^{7} + 23466090 T^{8} - 91017590 T^{9} + 758047824 T^{10} - 2714096862 T^{11} + 758047824 p T^{12} - 91017590 p^{2} T^{13} + 23466090 p^{3} T^{14} - 2695188 p^{4} T^{15} + 608249 p^{5} T^{16} - 63513 p^{6} T^{17} + 12166 p^{7} T^{18} - 1053 p^{8} T^{19} + 162 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 12 T + 164 T^{2} - 1556 T^{3} + 15616 T^{4} - 124012 T^{5} + 1013895 T^{6} - 7040784 T^{7} + 49839566 T^{8} - 309293704 T^{9} + 1931687428 T^{10} - 10680001976 T^{11} + 1931687428 p T^{12} - 309293704 p^{2} T^{13} + 49839566 p^{3} T^{14} - 7040784 p^{4} T^{15} + 1013895 p^{5} T^{16} - 124012 p^{6} T^{17} + 15616 p^{7} T^{18} - 1556 p^{8} T^{19} + 164 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - 19 T + 299 T^{2} - 3026 T^{3} + 29855 T^{4} - 232367 T^{5} + 1893333 T^{6} - 13007240 T^{7} + 97247462 T^{8} - 622220358 T^{9} + 4293909026 T^{10} - 25192306700 T^{11} + 4293909026 p T^{12} - 622220358 p^{2} T^{13} + 97247462 p^{3} T^{14} - 13007240 p^{4} T^{15} + 1893333 p^{5} T^{16} - 232367 p^{6} T^{17} + 29855 p^{7} T^{18} - 3026 p^{8} T^{19} + 299 p^{9} T^{20} - 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 - 9 T + 263 T^{2} - 1710 T^{3} + 29847 T^{4} - 133349 T^{5} + 1922353 T^{6} - 4465016 T^{7} + 77896422 T^{8} + 25594334 T^{9} + 2429925018 T^{10} + 6297341132 T^{11} + 2429925018 p T^{12} + 25594334 p^{2} T^{13} + 77896422 p^{3} T^{14} - 4465016 p^{4} T^{15} + 1922353 p^{5} T^{16} - 133349 p^{6} T^{17} + 29847 p^{7} T^{18} - 1710 p^{8} T^{19} + 263 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 - 15 T + 255 T^{2} - 2455 T^{3} + 27185 T^{4} - 217894 T^{5} + 1846022 T^{6} - 13249734 T^{7} + 98662127 T^{8} - 718191795 T^{9} + 4886685036 T^{10} - 34934321878 T^{11} + 4886685036 p T^{12} - 718191795 p^{2} T^{13} + 98662127 p^{3} T^{14} - 13249734 p^{4} T^{15} + 1846022 p^{5} T^{16} - 217894 p^{6} T^{17} + 27185 p^{7} T^{18} - 2455 p^{8} T^{19} + 255 p^{9} T^{20} - 15 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 - 35 T + 810 T^{2} - 14004 T^{3} + 207380 T^{4} - 2657861 T^{5} + 30622062 T^{6} - 317995297 T^{7} + 3039322164 T^{8} - 26684452634 T^{9} + 217536830827 T^{10} - 1640482517442 T^{11} + 217536830827 p T^{12} - 26684452634 p^{2} T^{13} + 3039322164 p^{3} T^{14} - 317995297 p^{4} T^{15} + 30622062 p^{5} T^{16} - 2657861 p^{6} T^{17} + 207380 p^{7} T^{18} - 14004 p^{8} T^{19} + 810 p^{9} T^{20} - 35 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 - 13 T + 355 T^{2} - 3917 T^{3} + 59667 T^{4} - 587600 T^{5} + 6624242 T^{6} - 60601636 T^{7} + 563071603 T^{8} - 4858682619 T^{9} + 39304507990 T^{10} - 316414839214 T^{11} + 39304507990 p T^{12} - 4858682619 p^{2} T^{13} + 563071603 p^{3} T^{14} - 60601636 p^{4} T^{15} + 6624242 p^{5} T^{16} - 587600 p^{6} T^{17} + 59667 p^{7} T^{18} - 3917 p^{8} T^{19} + 355 p^{9} T^{20} - 13 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 - 9 T + 422 T^{2} - 3261 T^{3} + 84106 T^{4} - 585093 T^{5} + 10888153 T^{6} - 70680988 T^{7} + 1048482778 T^{8} - 6384957130 T^{9} + 79640272008 T^{10} - 443464733902 T^{11} + 79640272008 p T^{12} - 6384957130 p^{2} T^{13} + 1048482778 p^{3} T^{14} - 70680988 p^{4} T^{15} + 10888153 p^{5} T^{16} - 585093 p^{6} T^{17} + 84106 p^{7} T^{18} - 3261 p^{8} T^{19} + 422 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 + T + 333 T^{2} + 638 T^{3} + 59387 T^{4} + 183285 T^{5} + 7347167 T^{6} + 30560872 T^{7} + 700322818 T^{8} + 3400020626 T^{9} + 55047335898 T^{10} + 268569881332 T^{11} + 55047335898 p T^{12} + 3400020626 p^{2} T^{13} + 700322818 p^{3} T^{14} + 30560872 p^{4} T^{15} + 7347167 p^{5} T^{16} + 183285 p^{6} T^{17} + 59387 p^{7} T^{18} + 638 p^{8} T^{19} + 333 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 - 14 T + 460 T^{2} - 4985 T^{3} + 97579 T^{4} - 889779 T^{5} + 13375484 T^{6} - 107292585 T^{7} + 1371734479 T^{8} - 10002402779 T^{9} + 114707718403 T^{10} - 772344883748 T^{11} + 114707718403 p T^{12} - 10002402779 p^{2} T^{13} + 1371734479 p^{3} T^{14} - 107292585 p^{4} T^{15} + 13375484 p^{5} T^{16} - 889779 p^{6} T^{17} + 97579 p^{7} T^{18} - 4985 p^{8} T^{19} + 460 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 + 20 T + 434 T^{2} + 6864 T^{3} + 101726 T^{4} + 1297304 T^{5} + 15856793 T^{6} + 173960008 T^{7} + 1848478870 T^{8} + 17939764980 T^{9} + 168530649416 T^{10} + 1469464726928 T^{11} + 168530649416 p T^{12} + 17939764980 p^{2} T^{13} + 1848478870 p^{3} T^{14} + 173960008 p^{4} T^{15} + 15856793 p^{5} T^{16} + 1297304 p^{6} T^{17} + 101726 p^{7} T^{18} + 6864 p^{8} T^{19} + 434 p^{9} T^{20} + 20 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 + 23 T + 736 T^{2} + 12240 T^{3} + 228094 T^{4} + 3009647 T^{5} + 42087186 T^{6} + 467303733 T^{7} + 5429720888 T^{8} + 52768209540 T^{9} + 536578013057 T^{10} + 4660295820994 T^{11} + 536578013057 p T^{12} + 52768209540 p^{2} T^{13} + 5429720888 p^{3} T^{14} + 467303733 p^{4} T^{15} + 42087186 p^{5} T^{16} + 3009647 p^{6} T^{17} + 228094 p^{7} T^{18} + 12240 p^{8} T^{19} + 736 p^{9} T^{20} + 23 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 12 T + 592 T^{2} - 6193 T^{3} + 164761 T^{4} - 1477529 T^{5} + 28827088 T^{6} - 221017883 T^{7} + 3620521637 T^{8} - 24209524583 T^{9} + 358500020515 T^{10} - 2169978265488 T^{11} + 358500020515 p T^{12} - 24209524583 p^{2} T^{13} + 3620521637 p^{3} T^{14} - 221017883 p^{4} T^{15} + 28827088 p^{5} T^{16} - 1477529 p^{6} T^{17} + 164761 p^{7} T^{18} - 6193 p^{8} T^{19} + 592 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 + T + 673 T^{2} + 1442 T^{3} + 221665 T^{4} + 632093 T^{5} + 47406969 T^{6} + 149040168 T^{7} + 7309766046 T^{8} + 22635956562 T^{9} + 849378499526 T^{10} + 2389655595756 T^{11} + 849378499526 p T^{12} + 22635956562 p^{2} T^{13} + 7309766046 p^{3} T^{14} + 149040168 p^{4} T^{15} + 47406969 p^{5} T^{16} + 632093 p^{6} T^{17} + 221665 p^{7} T^{18} + 1442 p^{8} T^{19} + 673 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 - 17 T + 733 T^{2} - 8806 T^{3} + 226517 T^{4} - 2124109 T^{5} + 43873857 T^{6} - 352763480 T^{7} + 6494674638 T^{8} - 47525276130 T^{9} + 783419884094 T^{10} - 5192510072708 T^{11} + 783419884094 p T^{12} - 47525276130 p^{2} T^{13} + 6494674638 p^{3} T^{14} - 352763480 p^{4} T^{15} + 43873857 p^{5} T^{16} - 2124109 p^{6} T^{17} + 226517 p^{7} T^{18} - 8806 p^{8} T^{19} + 733 p^{9} T^{20} - 17 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−2.86552155189184063624676730557, −2.78565398088123629936121986157, −2.66977325106842921194857942058, −2.36431116616049067607096201245, −2.35427986886721095932272631837, −2.26586785067618648048193091175, −2.26144993649695532002701087747, −2.18784018172780412778299987893, −2.12244228778325131825156845245, −1.95391077466816363893688843246, −1.93520104703070196093022463452, −1.81897835537519211401779871241, −1.79459521336370474249015982742, −1.69546046969106010655340865666, −1.61951553913237081356310170634, −1.39651778277447493487050539592, −1.25910007430027202389116555952, −1.25112824214554419542731816001, −1.07187627708111614033663742774, −1.02319833157380008975803505010, −0.953626228133436868857837721851, −0.919726262935713194546805769075, −0.857121845332315242430450258225, −0.72801842429990836773701507593, −0.70577492407685597189259077990, 0.70577492407685597189259077990, 0.72801842429990836773701507593, 0.857121845332315242430450258225, 0.919726262935713194546805769075, 0.953626228133436868857837721851, 1.02319833157380008975803505010, 1.07187627708111614033663742774, 1.25112824214554419542731816001, 1.25910007430027202389116555952, 1.39651778277447493487050539592, 1.61951553913237081356310170634, 1.69546046969106010655340865666, 1.79459521336370474249015982742, 1.81897835537519211401779871241, 1.93520104703070196093022463452, 1.95391077466816363893688843246, 2.12244228778325131825156845245, 2.18784018172780412778299987893, 2.26144993649695532002701087747, 2.26586785067618648048193091175, 2.35427986886721095932272631837, 2.36431116616049067607096201245, 2.66977325106842921194857942058, 2.78565398088123629936121986157, 2.86552155189184063624676730557
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.