Dirichlet series
| L(s) = 1 | − 11·2-s − 5·3-s + 66·4-s − 2·5-s + 55·6-s + 11·7-s − 286·8-s + 9-s + 22·10-s − 9·11-s − 330·12-s − 121·14-s + 10·15-s + 1.00e3·16-s − 11·17-s − 11·18-s + 11·19-s − 132·20-s − 55·21-s + 99·22-s − 8·23-s + 1.43e3·24-s − 17·25-s + 37·27-s + 726·28-s − 17·29-s − 110·30-s + ⋯ |
| L(s) = 1 | − 7.77·2-s − 2.88·3-s + 33·4-s − 0.894·5-s + 22.4·6-s + 4.15·7-s − 101.·8-s + 1/3·9-s + 6.95·10-s − 2.71·11-s − 95.2·12-s − 32.3·14-s + 2.58·15-s + 250.·16-s − 2.66·17-s − 2.59·18-s + 2.52·19-s − 29.5·20-s − 12.0·21-s + 21.1·22-s − 1.66·23-s + 291.·24-s − 3.39·25-s + 7.12·27-s + 137.·28-s − 3.15·29-s − 20.0·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 7^{11} \cdot 17^{11} \cdot 19^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 7^{11} \cdot 17^{11} \cdot 19^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{11} \cdot 7^{11} \cdot 17^{11} \cdot 19^{11}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.36045\times 10^{17}\) |
| Root analytic conductor: | \(6.00902\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(11\) |
| Selberg data: | \((22,\ 2^{11} \cdot 7^{11} \cdot 17^{11} \cdot 19^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )\) |
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 | \( ( 1 + T )^{11} \) |
| 7 | \( ( 1 - T )^{11} \) | |
| 17 | \( ( 1 + T )^{11} \) | |
| 19 | \( ( 1 - T )^{11} \) | |
| good | 3 | \( 1 + 5 T + 8 p T^{2} + 26 p T^{3} + 244 T^{4} + 71 p^{2} T^{5} + 539 p T^{6} + 3637 T^{7} + 7883 T^{8} + 5201 p T^{9} + 9929 p T^{10} + 52532 T^{11} + 9929 p^{2} T^{12} + 5201 p^{3} T^{13} + 7883 p^{3} T^{14} + 3637 p^{4} T^{15} + 539 p^{6} T^{16} + 71 p^{8} T^{17} + 244 p^{7} T^{18} + 26 p^{9} T^{19} + 8 p^{10} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) |
| 5 | \( 1 + 2 T + 21 T^{2} + 31 T^{3} + 192 T^{4} + 178 T^{5} + 207 p T^{6} + 357 T^{7} + 4348 T^{8} - 317 T^{9} + 787 p^{2} T^{10} - 3184 T^{11} + 787 p^{3} T^{12} - 317 p^{2} T^{13} + 4348 p^{3} T^{14} + 357 p^{4} T^{15} + 207 p^{6} T^{16} + 178 p^{6} T^{17} + 192 p^{7} T^{18} + 31 p^{8} T^{19} + 21 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 + 9 T + 84 T^{2} + 597 T^{3} + 3639 T^{4} + 20353 T^{5} + 100614 T^{6} + 463684 T^{7} + 178444 p T^{8} + 7717074 T^{9} + 28425520 T^{10} + 97165166 T^{11} + 28425520 p T^{12} + 7717074 p^{2} T^{13} + 178444 p^{4} T^{14} + 463684 p^{4} T^{15} + 100614 p^{5} T^{16} + 20353 p^{6} T^{17} + 3639 p^{7} T^{18} + 597 p^{8} T^{19} + 84 p^{9} T^{20} + 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 + 5 p T^{2} + 77 T^{3} + 2417 T^{4} + 3995 T^{5} + 64826 T^{6} + 120138 T^{7} + 1325867 T^{8} + 2468773 T^{9} + 21533860 T^{10} + 37095234 T^{11} + 21533860 p T^{12} + 2468773 p^{2} T^{13} + 1325867 p^{3} T^{14} + 120138 p^{4} T^{15} + 64826 p^{5} T^{16} + 3995 p^{6} T^{17} + 2417 p^{7} T^{18} + 77 p^{8} T^{19} + 5 p^{10} T^{20} + p^{11} T^{22} \) | |
| 23 | \( 1 + 8 T + 73 T^{2} + 260 T^{3} + 2105 T^{4} + 6226 T^{5} + 67107 T^{6} + 201695 T^{7} + 1998832 T^{8} + 4630870 T^{9} + 46127702 T^{10} + 96347738 T^{11} + 46127702 p T^{12} + 4630870 p^{2} T^{13} + 1998832 p^{3} T^{14} + 201695 p^{4} T^{15} + 67107 p^{5} T^{16} + 6226 p^{6} T^{17} + 2105 p^{7} T^{18} + 260 p^{8} T^{19} + 73 p^{9} T^{20} + 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 + 17 T + 336 T^{2} + 3800 T^{3} + 44418 T^{4} + 381951 T^{5} + 3330415 T^{6} + 23335979 T^{7} + 166646041 T^{8} + 998046343 T^{9} + 6164073689 T^{10} + 32588387260 T^{11} + 6164073689 p T^{12} + 998046343 p^{2} T^{13} + 166646041 p^{3} T^{14} + 23335979 p^{4} T^{15} + 3330415 p^{5} T^{16} + 381951 p^{6} T^{17} + 44418 p^{7} T^{18} + 3800 p^{8} T^{19} + 336 p^{9} T^{20} + 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 + 9 T + 127 T^{2} + 398 T^{3} + 4660 T^{4} + 616 T^{5} + 184887 T^{6} - 78925 T^{7} + 7825142 T^{8} - 3171467 T^{9} + 315534509 T^{10} + 85769554 T^{11} + 315534509 p T^{12} - 3171467 p^{2} T^{13} + 7825142 p^{3} T^{14} - 78925 p^{4} T^{15} + 184887 p^{5} T^{16} + 616 p^{6} T^{17} + 4660 p^{7} T^{18} + 398 p^{8} T^{19} + 127 p^{9} T^{20} + 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 + 14 T + 262 T^{2} + 3000 T^{3} + 35248 T^{4} + 333469 T^{5} + 3070446 T^{6} + 24900560 T^{7} + 193275759 T^{8} + 1367112181 T^{9} + 9231647564 T^{10} + 1553957888 p T^{11} + 9231647564 p T^{12} + 1367112181 p^{2} T^{13} + 193275759 p^{3} T^{14} + 24900560 p^{4} T^{15} + 3070446 p^{5} T^{16} + 333469 p^{6} T^{17} + 35248 p^{7} T^{18} + 3000 p^{8} T^{19} + 262 p^{9} T^{20} + 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 + 16 T + 303 T^{2} + 3156 T^{3} + 37200 T^{4} + 310769 T^{5} + 2993836 T^{6} + 22099932 T^{7} + 186688275 T^{8} + 1235975247 T^{9} + 9304211441 T^{10} + 55762862976 T^{11} + 9304211441 p T^{12} + 1235975247 p^{2} T^{13} + 186688275 p^{3} T^{14} + 22099932 p^{4} T^{15} + 2993836 p^{5} T^{16} + 310769 p^{6} T^{17} + 37200 p^{7} T^{18} + 3156 p^{8} T^{19} + 303 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 - 3 T + 227 T^{2} - 552 T^{3} + 24373 T^{4} - 54136 T^{5} + 1759547 T^{6} - 4192690 T^{7} + 102911358 T^{8} - 274186181 T^{9} + 5202124118 T^{10} - 13776397772 T^{11} + 5202124118 p T^{12} - 274186181 p^{2} T^{13} + 102911358 p^{3} T^{14} - 4192690 p^{4} T^{15} + 1759547 p^{5} T^{16} - 54136 p^{6} T^{17} + 24373 p^{7} T^{18} - 552 p^{8} T^{19} + 227 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 + 12 T + 363 T^{2} + 4185 T^{3} + 67278 T^{4} + 688858 T^{5} + 173179 p T^{6} + 72025117 T^{7} + 694391888 T^{8} + 5333525007 T^{9} + 43485347043 T^{10} + 290844346358 T^{11} + 43485347043 p T^{12} + 5333525007 p^{2} T^{13} + 694391888 p^{3} T^{14} + 72025117 p^{4} T^{15} + 173179 p^{6} T^{16} + 688858 p^{6} T^{17} + 67278 p^{7} T^{18} + 4185 p^{8} T^{19} + 363 p^{9} T^{20} + 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 + 34 T + 931 T^{2} + 17616 T^{3} + 288540 T^{4} + 3917311 T^{5} + 47762272 T^{6} + 9683438 p T^{7} + 5053650683 T^{8} + 45059426423 T^{9} + 371908716197 T^{10} + 2809836336308 T^{11} + 371908716197 p T^{12} + 45059426423 p^{2} T^{13} + 5053650683 p^{3} T^{14} + 9683438 p^{5} T^{15} + 47762272 p^{5} T^{16} + 3917311 p^{6} T^{17} + 288540 p^{7} T^{18} + 17616 p^{8} T^{19} + 931 p^{9} T^{20} + 34 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 + 23 T + 545 T^{2} + 8896 T^{3} + 135531 T^{4} + 1741614 T^{5} + 20830639 T^{6} + 222402230 T^{7} + 2226926362 T^{8} + 20328563547 T^{9} + 174855431438 T^{10} + 1382646742388 T^{11} + 174855431438 p T^{12} + 20328563547 p^{2} T^{13} + 2226926362 p^{3} T^{14} + 222402230 p^{4} T^{15} + 20830639 p^{5} T^{16} + 1741614 p^{6} T^{17} + 135531 p^{7} T^{18} + 8896 p^{8} T^{19} + 545 p^{9} T^{20} + 23 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 - 12 T + 299 T^{2} - 2541 T^{3} + 41582 T^{4} - 238260 T^{5} + 3179089 T^{6} - 8415399 T^{7} + 135608938 T^{8} + 449345075 T^{9} + 2702343051 T^{10} + 58880252016 T^{11} + 2702343051 p T^{12} + 449345075 p^{2} T^{13} + 135608938 p^{3} T^{14} - 8415399 p^{4} T^{15} + 3179089 p^{5} T^{16} - 238260 p^{6} T^{17} + 41582 p^{7} T^{18} - 2541 p^{8} T^{19} + 299 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 + 299 T^{2} - 806 T^{3} + 42355 T^{4} - 242586 T^{5} + 4175151 T^{6} - 33544767 T^{7} + 362531996 T^{8} - 2994615226 T^{9} + 28763614150 T^{10} - 213517140110 T^{11} + 28763614150 p T^{12} - 2994615226 p^{2} T^{13} + 362531996 p^{3} T^{14} - 33544767 p^{4} T^{15} + 4175151 p^{5} T^{16} - 242586 p^{6} T^{17} + 42355 p^{7} T^{18} - 806 p^{8} T^{19} + 299 p^{9} T^{20} + p^{11} T^{22} \) | |
| 71 | \( 1 + 42 T + 1442 T^{2} + 34247 T^{3} + 705320 T^{4} + 11968703 T^{5} + 181531289 T^{6} + 2397394916 T^{7} + 28753153240 T^{8} + 307289498631 T^{9} + 3006071749822 T^{10} + 26443041369546 T^{11} + 3006071749822 p T^{12} + 307289498631 p^{2} T^{13} + 28753153240 p^{3} T^{14} + 2397394916 p^{4} T^{15} + 181531289 p^{5} T^{16} + 11968703 p^{6} T^{17} + 705320 p^{7} T^{18} + 34247 p^{8} T^{19} + 1442 p^{9} T^{20} + 42 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 - 14 T + 535 T^{2} - 6555 T^{3} + 139313 T^{4} - 1530327 T^{5} + 23691868 T^{6} - 234527896 T^{7} + 2928667533 T^{8} - 25990504507 T^{9} + 275526452838 T^{10} - 2171407444346 T^{11} + 275526452838 p T^{12} - 25990504507 p^{2} T^{13} + 2928667533 p^{3} T^{14} - 234527896 p^{4} T^{15} + 23691868 p^{5} T^{16} - 1530327 p^{6} T^{17} + 139313 p^{7} T^{18} - 6555 p^{8} T^{19} + 535 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 + 7 T + 513 T^{2} + 3354 T^{3} + 138381 T^{4} + 836690 T^{5} + 24973831 T^{6} + 138546308 T^{7} + 3309206046 T^{8} + 16630914823 T^{9} + 335660351556 T^{10} + 1505624327876 T^{11} + 335660351556 p T^{12} + 16630914823 p^{2} T^{13} + 3309206046 p^{3} T^{14} + 138546308 p^{4} T^{15} + 24973831 p^{5} T^{16} + 836690 p^{6} T^{17} + 138381 p^{7} T^{18} + 3354 p^{8} T^{19} + 513 p^{9} T^{20} + 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - T + 593 T^{2} + 274 T^{3} + 166474 T^{4} + 322828 T^{5} + 29599535 T^{6} + 97221449 T^{7} + 3800706434 T^{8} + 15935651549 T^{9} + 383121252233 T^{10} + 1641063543434 T^{11} + 383121252233 p T^{12} + 15935651549 p^{2} T^{13} + 3800706434 p^{3} T^{14} + 97221449 p^{4} T^{15} + 29599535 p^{5} T^{16} + 322828 p^{6} T^{17} + 166474 p^{7} T^{18} + 274 p^{8} T^{19} + 593 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 + 6 T + 223 T^{2} + 687 T^{3} + 38060 T^{4} + 18498 T^{5} + 4493971 T^{6} - 1323775 T^{7} + 557943646 T^{8} - 308258385 T^{9} + 58535433831 T^{10} + 6656252282 T^{11} + 58535433831 p T^{12} - 308258385 p^{2} T^{13} + 557943646 p^{3} T^{14} - 1323775 p^{4} T^{15} + 4493971 p^{5} T^{16} + 18498 p^{6} T^{17} + 38060 p^{7} T^{18} + 687 p^{8} T^{19} + 223 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 + 15 T + 420 T^{2} + 4548 T^{3} + 82998 T^{4} + 675259 T^{5} + 11090827 T^{6} + 78697504 T^{7} + 1357626212 T^{8} + 9708999158 T^{9} + 160042652758 T^{10} + 1073355738168 T^{11} + 160042652758 p T^{12} + 9708999158 p^{2} T^{13} + 1357626212 p^{3} T^{14} + 78697504 p^{4} T^{15} + 11090827 p^{5} T^{16} + 675259 p^{6} T^{17} + 82998 p^{7} T^{18} + 4548 p^{8} T^{19} + 420 p^{9} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−3.03664900477361027301485223562, −3.01318323773660523478441488563, −2.95281486615821891555851464425, −2.47607184041084852449518197752, −2.46837810291462010965848475000, −2.43942677710487864695384907639, −2.40722960447330907450344815027, −2.40435890384507627681775396461, −2.29792625988141520639277077264, −2.12303420525796562304675060997, −2.11331515568756504082746809222, −2.05212841741910440938997971458, −2.04621904406121101780555636449, −1.88107716680098884878120589939, −1.66482179601354161550584720002, −1.63003297287088047263265040476, −1.51733527680519471261133040749, −1.46166068290410640017390698792, −1.34467360369072102358449416123, −1.32089629311397761009525882273, −1.24166903487153794748230957286, −1.16958600389793511237118923026, −1.04258464273113791740777943368, −0.980339301576717835105851412321, −0.825925040010548811394814691822, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.825925040010548811394814691822, 0.980339301576717835105851412321, 1.04258464273113791740777943368, 1.16958600389793511237118923026, 1.24166903487153794748230957286, 1.32089629311397761009525882273, 1.34467360369072102358449416123, 1.46166068290410640017390698792, 1.51733527680519471261133040749, 1.63003297287088047263265040476, 1.66482179601354161550584720002, 1.88107716680098884878120589939, 2.04621904406121101780555636449, 2.05212841741910440938997971458, 2.11331515568756504082746809222, 2.12303420525796562304675060997, 2.29792625988141520639277077264, 2.40435890384507627681775396461, 2.40722960447330907450344815027, 2.43942677710487864695384907639, 2.46837810291462010965848475000, 2.47607184041084852449518197752, 2.95281486615821891555851464425, 3.01318323773660523478441488563, 3.03664900477361027301485223562
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.