Dirichlet series
| L(s) = 1 | − 11·2-s − 4·3-s + 66·4-s + 5-s + 44·6-s − 6·7-s − 286·8-s − 7·9-s − 11·10-s + 4·11-s − 264·12-s − 19·13-s + 66·14-s − 4·15-s + 1.00e3·16-s − 8·17-s + 77·18-s − 9·19-s + 66·20-s + 24·21-s − 44·22-s + 2·23-s + 1.14e3·24-s − 28·25-s + 209·26-s + 48·27-s − 396·28-s + ⋯ |
| L(s) = 1 | − 7.77·2-s − 2.30·3-s + 33·4-s + 0.447·5-s + 17.9·6-s − 2.26·7-s − 101.·8-s − 7/3·9-s − 3.47·10-s + 1.20·11-s − 76.2·12-s − 5.26·13-s + 17.6·14-s − 1.03·15-s + 250.·16-s − 1.94·17-s + 18.1·18-s − 2.06·19-s + 14.7·20-s + 5.23·21-s − 9.38·22-s + 0.417·23-s + 233.·24-s − 5.59·25-s + 40.9·26-s + 9.23·27-s − 74.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 751^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 751^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{11} \cdot 751^{11}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(7.38613\times 10^{11}\) |
| Root analytic conductor: | \(3.46316\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(11\) |
| Selberg data: | \((22,\ 2^{11} \cdot 751^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{11} \) |
| 751 | \( ( 1 + T )^{11} \) | |
| good | 3 | \( 1 + 4 T + 23 T^{2} + 8 p^{2} T^{3} + 28 p^{2} T^{4} + 218 p T^{5} + 1771 T^{6} + 3959 T^{7} + 2998 p T^{8} + 1961 p^{2} T^{9} + 11590 p T^{10} + 60227 T^{11} + 11590 p^{2} T^{12} + 1961 p^{4} T^{13} + 2998 p^{4} T^{14} + 3959 p^{4} T^{15} + 1771 p^{5} T^{16} + 218 p^{7} T^{17} + 28 p^{9} T^{18} + 8 p^{10} T^{19} + 23 p^{9} T^{20} + 4 p^{10} T^{21} + p^{11} T^{22} \) |
| 5 | \( 1 - T + 29 T^{2} - 4 p T^{3} + 394 T^{4} - 94 T^{5} + 3288 T^{6} + 1187 T^{7} + 19311 T^{8} + 20469 T^{9} + 94521 T^{10} + 140607 T^{11} + 94521 p T^{12} + 20469 p^{2} T^{13} + 19311 p^{3} T^{14} + 1187 p^{4} T^{15} + 3288 p^{5} T^{16} - 94 p^{6} T^{17} + 394 p^{7} T^{18} - 4 p^{9} T^{19} + 29 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 7 | \( 1 + 6 T + 59 T^{2} + 260 T^{3} + 209 p T^{4} + 5154 T^{5} + 21418 T^{6} + 9103 p T^{7} + 218259 T^{8} + 576308 T^{9} + 251186 p T^{10} + 4312446 T^{11} + 251186 p^{2} T^{12} + 576308 p^{2} T^{13} + 218259 p^{3} T^{14} + 9103 p^{5} T^{15} + 21418 p^{5} T^{16} + 5154 p^{6} T^{17} + 209 p^{8} T^{18} + 260 p^{8} T^{19} + 59 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 - 4 T + 76 T^{2} - 218 T^{3} + 2543 T^{4} - 5171 T^{5} + 52856 T^{6} - 76251 T^{7} + 824629 T^{8} - 903598 T^{9} + 10686030 T^{10} - 10096417 T^{11} + 10686030 p T^{12} - 903598 p^{2} T^{13} + 824629 p^{3} T^{14} - 76251 p^{4} T^{15} + 52856 p^{5} T^{16} - 5171 p^{6} T^{17} + 2543 p^{7} T^{18} - 218 p^{8} T^{19} + 76 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 + 19 T + 251 T^{2} + 2486 T^{3} + 20498 T^{4} + 144288 T^{5} + 895242 T^{6} + 4944973 T^{7} + 24659831 T^{8} + 111493297 T^{9} + 35388967 p T^{10} + 1733530095 T^{11} + 35388967 p^{2} T^{12} + 111493297 p^{2} T^{13} + 24659831 p^{3} T^{14} + 4944973 p^{4} T^{15} + 895242 p^{5} T^{16} + 144288 p^{6} T^{17} + 20498 p^{7} T^{18} + 2486 p^{8} T^{19} + 251 p^{9} T^{20} + 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 + 8 T + 120 T^{2} + 743 T^{3} + 6373 T^{4} + 30365 T^{5} + 192554 T^{6} + 708204 T^{7} + 3751031 T^{8} + 10892706 T^{9} + 57642117 T^{10} + 157981491 T^{11} + 57642117 p T^{12} + 10892706 p^{2} T^{13} + 3751031 p^{3} T^{14} + 708204 p^{4} T^{15} + 192554 p^{5} T^{16} + 30365 p^{6} T^{17} + 6373 p^{7} T^{18} + 743 p^{8} T^{19} + 120 p^{9} T^{20} + 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 + 9 T + 132 T^{2} + 866 T^{3} + 7833 T^{4} + 42985 T^{5} + 304739 T^{6} + 1479206 T^{7} + 8926075 T^{8} + 39204165 T^{9} + 208798205 T^{10} + 830509179 T^{11} + 208798205 p T^{12} + 39204165 p^{2} T^{13} + 8926075 p^{3} T^{14} + 1479206 p^{4} T^{15} + 304739 p^{5} T^{16} + 42985 p^{6} T^{17} + 7833 p^{7} T^{18} + 866 p^{8} T^{19} + 132 p^{9} T^{20} + 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 2 T + 116 T^{2} - 199 T^{3} + 6875 T^{4} - 8589 T^{5} + 279111 T^{6} - 206919 T^{7} + 380868 p T^{8} - 3090421 T^{9} + 229603101 T^{10} - 46207660 T^{11} + 229603101 p T^{12} - 3090421 p^{2} T^{13} + 380868 p^{4} T^{14} - 206919 p^{4} T^{15} + 279111 p^{5} T^{16} - 8589 p^{6} T^{17} + 6875 p^{7} T^{18} - 199 p^{8} T^{19} + 116 p^{9} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 - 13 T + 224 T^{2} - 2354 T^{3} + 25468 T^{4} - 217411 T^{5} + 1843417 T^{6} - 13336997 T^{7} + 94415573 T^{8} - 591192671 T^{9} + 3604728408 T^{10} - 19668193409 T^{11} + 3604728408 p T^{12} - 591192671 p^{2} T^{13} + 94415573 p^{3} T^{14} - 13336997 p^{4} T^{15} + 1843417 p^{5} T^{16} - 217411 p^{6} T^{17} + 25468 p^{7} T^{18} - 2354 p^{8} T^{19} + 224 p^{9} T^{20} - 13 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 + 19 T + 368 T^{2} + 4447 T^{3} + 51938 T^{4} + 471895 T^{5} + 4148528 T^{6} + 30640520 T^{7} + 221593872 T^{8} + 1398891383 T^{9} + 8763296259 T^{10} + 48830151971 T^{11} + 8763296259 p T^{12} + 1398891383 p^{2} T^{13} + 221593872 p^{3} T^{14} + 30640520 p^{4} T^{15} + 4148528 p^{5} T^{16} + 471895 p^{6} T^{17} + 51938 p^{7} T^{18} + 4447 p^{8} T^{19} + 368 p^{9} T^{20} + 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 + 29 T + 530 T^{2} + 7415 T^{3} + 87962 T^{4} + 906024 T^{5} + 8377271 T^{6} + 70517085 T^{7} + 547365646 T^{8} + 3935903852 T^{9} + 26446088879 T^{10} + 166181713355 T^{11} + 26446088879 p T^{12} + 3935903852 p^{2} T^{13} + 547365646 p^{3} T^{14} + 70517085 p^{4} T^{15} + 8377271 p^{5} T^{16} + 906024 p^{6} T^{17} + 87962 p^{7} T^{18} + 7415 p^{8} T^{19} + 530 p^{9} T^{20} + 29 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 + 23 T + 524 T^{2} + 7801 T^{3} + 110137 T^{4} + 1250171 T^{5} + 13409888 T^{6} + 123776224 T^{7} + 1081864063 T^{8} + 8371077407 T^{9} + 61452377952 T^{10} + 403835689705 T^{11} + 61452377952 p T^{12} + 8371077407 p^{2} T^{13} + 1081864063 p^{3} T^{14} + 123776224 p^{4} T^{15} + 13409888 p^{5} T^{16} + 1250171 p^{6} T^{17} + 110137 p^{7} T^{18} + 7801 p^{8} T^{19} + 524 p^{9} T^{20} + 23 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 + 13 T + 238 T^{2} + 2106 T^{3} + 25816 T^{4} + 201269 T^{5} + 2028245 T^{6} + 14214483 T^{7} + 122945681 T^{8} + 793330591 T^{9} + 6220241242 T^{10} + 37312391727 T^{11} + 6220241242 p T^{12} + 793330591 p^{2} T^{13} + 122945681 p^{3} T^{14} + 14214483 p^{4} T^{15} + 2028245 p^{5} T^{16} + 201269 p^{6} T^{17} + 25816 p^{7} T^{18} + 2106 p^{8} T^{19} + 238 p^{9} T^{20} + 13 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 + 16 T + 321 T^{2} + 3647 T^{3} + 48416 T^{4} + 446650 T^{5} + 4726823 T^{6} + 38156849 T^{7} + 349654826 T^{8} + 2519117392 T^{9} + 20488582878 T^{10} + 132762074369 T^{11} + 20488582878 p T^{12} + 2519117392 p^{2} T^{13} + 349654826 p^{3} T^{14} + 38156849 p^{4} T^{15} + 4726823 p^{5} T^{16} + 446650 p^{6} T^{17} + 48416 p^{7} T^{18} + 3647 p^{8} T^{19} + 321 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 + 25 T + 658 T^{2} + 10501 T^{3} + 166508 T^{4} + 2024488 T^{5} + 24321735 T^{6} + 243039987 T^{7} + 2399991584 T^{8} + 20406299116 T^{9} + 3233613123 p T^{10} + 1255661569249 T^{11} + 3233613123 p^{2} T^{12} + 20406299116 p^{2} T^{13} + 2399991584 p^{3} T^{14} + 243039987 p^{4} T^{15} + 24321735 p^{5} T^{16} + 2024488 p^{6} T^{17} + 166508 p^{7} T^{18} + 10501 p^{8} T^{19} + 658 p^{9} T^{20} + 25 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 - 6 T + 233 T^{2} - 376 T^{3} + 19318 T^{4} + 21737 T^{5} + 1369125 T^{6} - 1912461 T^{7} + 100132738 T^{8} - 250397075 T^{9} + 3792758135 T^{10} - 6632269702 T^{11} + 3792758135 p T^{12} - 250397075 p^{2} T^{13} + 100132738 p^{3} T^{14} - 1912461 p^{4} T^{15} + 1369125 p^{5} T^{16} + 21737 p^{6} T^{17} + 19318 p^{7} T^{18} - 376 p^{8} T^{19} + 233 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 - 10 T + 516 T^{2} - 4272 T^{3} + 119700 T^{4} - 822722 T^{5} + 16776452 T^{6} - 96335141 T^{7} + 1633182154 T^{8} - 8007968175 T^{9} + 121923996153 T^{10} - 532123144541 T^{11} + 121923996153 p T^{12} - 8007968175 p^{2} T^{13} + 1633182154 p^{3} T^{14} - 96335141 p^{4} T^{15} + 16776452 p^{5} T^{16} - 822722 p^{6} T^{17} + 119700 p^{7} T^{18} - 4272 p^{8} T^{19} + 516 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 + 16 T + 615 T^{2} + 8234 T^{3} + 175581 T^{4} + 1993737 T^{5} + 30895766 T^{6} + 301684551 T^{7} + 3757993440 T^{8} + 31868672758 T^{9} + 334867476598 T^{10} + 2472521226173 T^{11} + 334867476598 p T^{12} + 31868672758 p^{2} T^{13} + 3757993440 p^{3} T^{14} + 301684551 p^{4} T^{15} + 30895766 p^{5} T^{16} + 1993737 p^{6} T^{17} + 175581 p^{7} T^{18} + 8234 p^{8} T^{19} + 615 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 - 8 T + 368 T^{2} - 2391 T^{3} + 63944 T^{4} - 317254 T^{5} + 6876058 T^{6} - 22215854 T^{7} + 519562092 T^{8} - 718721756 T^{9} + 33189514387 T^{10} - 12488035445 T^{11} + 33189514387 p T^{12} - 718721756 p^{2} T^{13} + 519562092 p^{3} T^{14} - 22215854 p^{4} T^{15} + 6876058 p^{5} T^{16} - 317254 p^{6} T^{17} + 63944 p^{7} T^{18} - 2391 p^{8} T^{19} + 368 p^{9} T^{20} - 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 + 56 T + 1803 T^{2} + 40850 T^{3} + 728151 T^{4} + 10801497 T^{5} + 139563167 T^{6} + 1620820864 T^{7} + 17335614757 T^{8} + 173046152513 T^{9} + 1622262774152 T^{10} + 14295122392183 T^{11} + 1622262774152 p T^{12} + 173046152513 p^{2} T^{13} + 17335614757 p^{3} T^{14} + 1620820864 p^{4} T^{15} + 139563167 p^{5} T^{16} + 10801497 p^{6} T^{17} + 728151 p^{7} T^{18} + 40850 p^{8} T^{19} + 1803 p^{9} T^{20} + 56 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 2 T + 351 T^{2} - 590 T^{3} + 66579 T^{4} - 200805 T^{5} + 8944941 T^{6} - 40985072 T^{7} + 914532221 T^{8} - 5490048169 T^{9} + 80248761640 T^{10} - 521211788815 T^{11} + 80248761640 p T^{12} - 5490048169 p^{2} T^{13} + 914532221 p^{3} T^{14} - 40985072 p^{4} T^{15} + 8944941 p^{5} T^{16} - 200805 p^{6} T^{17} + 66579 p^{7} T^{18} - 590 p^{8} T^{19} + 351 p^{9} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 21 T + 861 T^{2} - 13395 T^{3} + 311794 T^{4} - 3842050 T^{5} + 65761741 T^{6} - 670200108 T^{7} + 9377948350 T^{8} - 81917480725 T^{9} + 993476580480 T^{10} - 7651988189753 T^{11} + 993476580480 p T^{12} - 81917480725 p^{2} T^{13} + 9377948350 p^{3} T^{14} - 670200108 p^{4} T^{15} + 65761741 p^{5} T^{16} - 3842050 p^{6} T^{17} + 311794 p^{7} T^{18} - 13395 p^{8} T^{19} + 861 p^{9} T^{20} - 21 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 + 24 T + 730 T^{2} + 11824 T^{3} + 220808 T^{4} + 2841686 T^{5} + 41137351 T^{6} + 447517393 T^{7} + 5494330029 T^{8} + 52813110330 T^{9} + 581943235662 T^{10} + 5109080651615 T^{11} + 581943235662 p T^{12} + 52813110330 p^{2} T^{13} + 5494330029 p^{3} T^{14} + 447517393 p^{4} T^{15} + 41137351 p^{5} T^{16} + 2841686 p^{6} T^{17} + 220808 p^{7} T^{18} + 11824 p^{8} T^{19} + 730 p^{9} T^{20} + 24 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 + 84 T + 3898 T^{2} + 126840 T^{3} + 3203709 T^{4} + 66248405 T^{5} + 1161441294 T^{6} + 17698825499 T^{7} + 239157752199 T^{8} + 2912808637566 T^{9} + 32397739228254 T^{10} + 331934933022227 T^{11} + 32397739228254 p T^{12} + 2912808637566 p^{2} T^{13} + 239157752199 p^{3} T^{14} + 17698825499 p^{4} T^{15} + 1161441294 p^{5} T^{16} + 66248405 p^{6} T^{17} + 3203709 p^{7} T^{18} + 126840 p^{8} T^{19} + 3898 p^{9} T^{20} + 84 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−3.53875215935714061926334580474, −3.46832275144324940258307867382, −3.46553788247114735921634116654, −3.17543721925371929513680070196, −3.11295197640407919981755569129, −2.94242333830367144346149977731, −2.80944296589361881929061826655, −2.75677683343591506794633609296, −2.66691331403447115272984596324, −2.65574415095143538396679161077, −2.64900716782881168749510961510, −2.58084737601136772017862124147, −2.55984113148384415249592487526, −2.21703251142069649081973264088, −2.00046676742539989282627255128, −1.97849624922464241044315344617, −1.85931233632875258013399521483, −1.84409985483334531283121799001, −1.71219486929232588262523554336, −1.67238335714041377558229799328, −1.50600127946991471987917732394, −1.45474014450449100931610179925, −1.42574509890387686696723514415, −1.14657240717087337933250302371, −0.967422723558437876828045005312, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.967422723558437876828045005312, 1.14657240717087337933250302371, 1.42574509890387686696723514415, 1.45474014450449100931610179925, 1.50600127946991471987917732394, 1.67238335714041377558229799328, 1.71219486929232588262523554336, 1.84409985483334531283121799001, 1.85931233632875258013399521483, 1.97849624922464241044315344617, 2.00046676742539989282627255128, 2.21703251142069649081973264088, 2.55984113148384415249592487526, 2.58084737601136772017862124147, 2.64900716782881168749510961510, 2.65574415095143538396679161077, 2.66691331403447115272984596324, 2.75677683343591506794633609296, 2.80944296589361881929061826655, 2.94242333830367144346149977731, 3.11295197640407919981755569129, 3.17543721925371929513680070196, 3.46553788247114735921634116654, 3.46832275144324940258307867382, 3.53875215935714061926334580474
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.