Dirichlet series
| L(s) = 1 | − 10·3-s − 4·4-s − 6·7-s + 36·9-s + 40·12-s + 42·13-s + 4·16-s + 134·19-s + 60·21-s + 17·25-s + 24·28-s + 124·31-s − 144·36-s + 90·37-s − 420·39-s + 156·43-s − 40·48-s + 101·49-s − 168·52-s − 1.34e3·57-s + 126·61-s − 216·63-s − 14·64-s + 368·67-s − 24·73-s − 170·75-s − 536·76-s + ⋯ |
| L(s) = 1 | − 3.33·3-s − 4-s − 6/7·7-s + 4·9-s + 10/3·12-s + 3.23·13-s + 1/4·16-s + 7.05·19-s + 20/7·21-s + 0.679·25-s + 6/7·28-s + 4·31-s − 4·36-s + 2.43·37-s − 10.7·39-s + 3.62·43-s − 5/6·48-s + 2.06·49-s − 3.23·52-s − 23.5·57-s + 2.06·61-s − 3.42·63-s − 0.218·64-s + 5.49·67-s − 0.328·73-s − 2.26·75-s − 7.05·76-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 11^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.39202\times 10^{15}\) |
| Root analytic conductor: | \(3.14500\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 11^{32} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(42.59360068\) |
| \(L(\frac12)\) | \(\approx\) | \(42.59360068\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 10 T + 64 T^{2} + 280 T^{3} + 1045 T^{4} + 130 p^{3} T^{5} + 428 p^{3} T^{6} + 1360 p^{3} T^{7} + 463 p^{5} T^{8} + 1360 p^{5} T^{9} + 428 p^{7} T^{10} + 130 p^{9} T^{11} + 1045 p^{8} T^{12} + 280 p^{10} T^{13} + 64 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + p^{2} T^{2} + 3 p^{2} T^{4} + 23 p T^{6} + 331 T^{8} + 785 p T^{10} + 5163 T^{12} + 12695 T^{14} + 94057 T^{16} + 12695 p^{4} T^{18} + 5163 p^{8} T^{20} + 785 p^{13} T^{22} + 331 p^{16} T^{24} + 23 p^{21} T^{26} + 3 p^{26} T^{28} + p^{30} T^{30} + p^{32} T^{32} \) |
| 5 | \( 1 - 17 T^{2} + 486 p T^{4} - 10832 T^{6} + 2625694 T^{8} + 7572877 T^{10} + 459500103 p T^{12} + 7383221492 T^{14} + 1688830878256 T^{16} + 7383221492 p^{4} T^{18} + 459500103 p^{9} T^{20} + 7572877 p^{12} T^{22} + 2625694 p^{16} T^{24} - 10832 p^{20} T^{26} + 486 p^{25} T^{28} - 17 p^{28} T^{30} + p^{32} T^{32} \) | |
| 7 | \( ( 1 + 3 T - 37 T^{2} + 358 T^{3} + 478 p T^{4} - 4635 T^{5} + 54602 T^{6} + 1007520 T^{7} + 4844597 T^{8} + 1007520 p^{2} T^{9} + 54602 p^{4} T^{10} - 4635 p^{6} T^{11} + 478 p^{9} T^{12} + 358 p^{10} T^{13} - 37 p^{12} T^{14} + 3 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 21 T + 81 T^{2} + 770 T^{3} + 7728 T^{4} - 805195 T^{5} + 9940744 T^{6} - 7821066 T^{7} - 396624107 T^{8} - 7821066 p^{2} T^{9} + 9940744 p^{4} T^{10} - 805195 p^{6} T^{11} + 7728 p^{8} T^{12} + 770 p^{10} T^{13} + 81 p^{12} T^{14} - 21 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 + 894 T^{2} + 336120 T^{4} + 20399550 T^{6} - 26640872055 T^{8} - 10092273774498 T^{10} - 482862517622152 T^{12} + 723393848340773460 T^{14} + \)\(32\!\cdots\!25\)\( T^{16} + 723393848340773460 p^{4} T^{18} - 482862517622152 p^{8} T^{20} - 10092273774498 p^{12} T^{22} - 26640872055 p^{16} T^{24} + 20399550 p^{20} T^{26} + 336120 p^{24} T^{28} + 894 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 - 67 T + 1578 T^{2} - 12676 T^{3} - 81938 T^{4} + 6545735 T^{5} - 212459133 T^{6} + 123819472 p T^{7} - 11642360 p^{2} T^{8} + 123819472 p^{3} T^{9} - 212459133 p^{4} T^{10} + 6545735 p^{6} T^{11} - 81938 p^{8} T^{12} - 12676 p^{10} T^{13} + 1578 p^{12} T^{14} - 67 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 2384 T^{2} + 3158579 T^{4} - 2740573584 T^{6} + 1706195561800 T^{8} - 2740573584 p^{4} T^{10} + 3158579 p^{8} T^{12} - 2384 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 29 | \( 1 + 3112 T^{2} + 6365538 T^{4} + 10682263816 T^{6} + 15305712405667 T^{8} + 18918572757319240 T^{10} + 20690280210889005732 T^{12} + \)\(20\!\cdots\!92\)\( T^{14} + \)\(18\!\cdots\!65\)\( T^{16} + \)\(20\!\cdots\!92\)\( p^{4} T^{18} + 20690280210889005732 p^{8} T^{20} + 18918572757319240 p^{12} T^{22} + 15305712405667 p^{16} T^{24} + 10682263816 p^{20} T^{26} + 6365538 p^{24} T^{28} + 3112 p^{28} T^{30} + p^{32} T^{32} \) | |
| 31 | \( ( 1 - 2 p T + 2136 T^{2} - 51368 T^{3} + 1877341 T^{4} + 82834 p T^{5} - 81168276 p T^{6} + 116136241856 T^{7} - 2932487147363 T^{8} + 116136241856 p^{2} T^{9} - 81168276 p^{5} T^{10} + 82834 p^{7} T^{11} + 1877341 p^{8} T^{12} - 51368 p^{10} T^{13} + 2136 p^{12} T^{14} - 2 p^{15} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 45 T + 645 T^{2} - 16850 T^{3} - 1356006 T^{4} + 101768095 T^{5} + 1589698690 T^{6} - 185593158390 T^{7} + 6348820327681 T^{8} - 185593158390 p^{2} T^{9} + 1589698690 p^{4} T^{10} + 101768095 p^{6} T^{11} - 1356006 p^{8} T^{12} - 16850 p^{10} T^{13} + 645 p^{12} T^{14} - 45 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( 1 - 995 T^{2} + 1570671 T^{4} + 1044796720 T^{6} - 7793718391520 T^{8} + 15577888312588045 T^{10} + 8181601625193452394 T^{12} - \)\(48\!\cdots\!50\)\( T^{14} + \)\(11\!\cdots\!09\)\( T^{16} - \)\(48\!\cdots\!50\)\( p^{4} T^{18} + 8181601625193452394 p^{8} T^{20} + 15577888312588045 p^{12} T^{22} - 7793718391520 p^{16} T^{24} + 1044796720 p^{20} T^{26} + 1570671 p^{24} T^{28} - 995 p^{28} T^{30} + p^{32} T^{32} \) | |
| 43 | \( ( 1 - 39 T + 5868 T^{2} - 143114 T^{3} + 14177853 T^{4} - 143114 p^{2} T^{5} + 5868 p^{4} T^{6} - 39 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 47 | \( 1 - 995 T^{2} + 2709666 T^{4} - 5345087660 T^{6} + 2560879190710 T^{8} + 1540723914202015 T^{10} + 68482758794882002179 T^{12} - \)\(35\!\cdots\!80\)\( T^{14} + \)\(20\!\cdots\!64\)\( T^{16} - \)\(35\!\cdots\!80\)\( p^{4} T^{18} + 68482758794882002179 p^{8} T^{20} + 1540723914202015 p^{12} T^{22} + 2560879190710 p^{16} T^{24} - 5345087660 p^{20} T^{26} + 2709666 p^{24} T^{28} - 995 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 + 9862 T^{2} + 44250456 T^{4} + 83411867458 T^{6} - 136888230588719 T^{8} - 1037106678398694674 T^{10} - \)\(80\!\cdots\!96\)\( T^{12} + \)\(10\!\cdots\!24\)\( T^{14} + \)\(50\!\cdots\!97\)\( T^{16} + \)\(10\!\cdots\!24\)\( p^{4} T^{18} - \)\(80\!\cdots\!96\)\( p^{8} T^{20} - 1037106678398694674 p^{12} T^{22} - 136888230588719 p^{16} T^{24} + 83411867458 p^{20} T^{26} + 44250456 p^{24} T^{28} + 9862 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( 1 - 1959 T^{2} - 22685748 T^{4} + 35157453984 T^{6} + 163809750679236 T^{8} + 484303216945908915 T^{10} - \)\(24\!\cdots\!77\)\( T^{12} - \)\(76\!\cdots\!00\)\( T^{14} + \)\(50\!\cdots\!92\)\( T^{16} - \)\(76\!\cdots\!00\)\( p^{4} T^{18} - \)\(24\!\cdots\!77\)\( p^{8} T^{20} + 484303216945908915 p^{12} T^{22} + 163809750679236 p^{16} T^{24} + 35157453984 p^{20} T^{26} - 22685748 p^{24} T^{28} - 1959 p^{28} T^{30} + p^{32} T^{32} \) | |
| 61 | \( ( 1 - 63 T - 7114 T^{2} + 428758 T^{3} + 14002276 T^{4} - 1658235189 T^{5} + 137533071599 T^{6} + 2282091395694 T^{7} - 914529740532628 T^{8} + 2282091395694 p^{2} T^{9} + 137533071599 p^{4} T^{10} - 1658235189 p^{6} T^{11} + 14002276 p^{8} T^{12} + 428758 p^{10} T^{13} - 7114 p^{12} T^{14} - 63 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 92 T + 14029 T^{2} - 686396 T^{3} + 73672501 T^{4} - 686396 p^{2} T^{5} + 14029 p^{4} T^{6} - 92 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 71 | \( 1 + 18983 T^{2} + 212109414 T^{4} + 1794032145180 T^{6} + 12788451374810298 T^{8} + 81069844229226960885 T^{10} + \)\(47\!\cdots\!91\)\( T^{12} + \)\(25\!\cdots\!92\)\( T^{14} + \)\(13\!\cdots\!08\)\( T^{16} + \)\(25\!\cdots\!92\)\( p^{4} T^{18} + \)\(47\!\cdots\!91\)\( p^{8} T^{20} + 81069844229226960885 p^{12} T^{22} + 12788451374810298 p^{16} T^{24} + 1794032145180 p^{20} T^{26} + 212109414 p^{24} T^{28} + 18983 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 + 12 T - 4158 T^{2} - 690642 T^{3} + 34555347 T^{4} - 1402256700 T^{5} - 68554055572 T^{6} - 4759965946776 T^{7} + 2851625758685025 T^{8} - 4759965946776 p^{2} T^{9} - 68554055572 p^{4} T^{10} - 1402256700 p^{6} T^{11} + 34555347 p^{8} T^{12} - 690642 p^{10} T^{13} - 4158 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 157 T + 7773 T^{2} - 944146 T^{3} + 137198002 T^{4} - 10621769425 T^{5} + 1102248505962 T^{6} - 91369308370742 T^{7} + 5518958597130505 T^{8} - 91369308370742 p^{2} T^{9} + 1102248505962 p^{4} T^{10} - 10621769425 p^{6} T^{11} + 137198002 p^{8} T^{12} - 944146 p^{10} T^{13} + 7773 p^{12} T^{14} - 157 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( 1 + 28010 T^{2} + 425035818 T^{4} + 4516595468385 T^{6} + 39017282313118833 T^{8} + \)\(29\!\cdots\!60\)\( T^{10} + \)\(20\!\cdots\!81\)\( T^{12} + \)\(13\!\cdots\!25\)\( T^{14} + \)\(89\!\cdots\!30\)\( T^{16} + \)\(13\!\cdots\!25\)\( p^{4} T^{18} + \)\(20\!\cdots\!81\)\( p^{8} T^{20} + \)\(29\!\cdots\!60\)\( p^{12} T^{22} + 39017282313118833 p^{16} T^{24} + 4516595468385 p^{20} T^{26} + 425035818 p^{24} T^{28} + 28010 p^{28} T^{30} + p^{32} T^{32} \) | |
| 89 | \( ( 1 - 54839 T^{2} + 1370536886 T^{4} - 20394246717258 T^{6} + 197742529962478711 T^{8} - 20394246717258 p^{4} T^{10} + 1370536886 p^{8} T^{12} - 54839 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 36 T - 15693 T^{2} + 1693818 T^{3} + 119942457 T^{4} - 7626572010 T^{5} - 578486645527 T^{6} + 23274720379476 T^{7} + 13991351802208980 T^{8} + 23274720379476 p^{2} T^{9} - 578486645527 p^{4} T^{10} - 7626572010 p^{6} T^{11} + 119942457 p^{8} T^{12} + 1693818 p^{10} T^{13} - 15693 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.80768075655858865802505780177, −2.76298966005525945584519974841, −2.73415429501887836121790070821, −2.71802474960887502227126960641, −2.57542467649560812565451861393, −2.49741506707946857156691976439, −2.44298193785771087230921645348, −2.23985211080503827597293540726, −2.19164304065141788144265792194, −2.03612154319808083085572565231, −1.82784581193390095900187224554, −1.71405545678666072810966990178, −1.70814922313784356994346333417, −1.45202324460079378358063871590, −1.32448698846827370000132225603, −0.981037062714141615566084749747, −0.945066903836358747274923995209, −0.921206915201657954834574957219, −0.898417773082125678666154970933, −0.860976019529524374877507430916, −0.69796084898117227907664335517, −0.64331446711368242763070139141, −0.61913438739508660115285473625, −0.59638576433340924846668735860, −0.32670212836539773344081153887, 0.32670212836539773344081153887, 0.59638576433340924846668735860, 0.61913438739508660115285473625, 0.64331446711368242763070139141, 0.69796084898117227907664335517, 0.860976019529524374877507430916, 0.898417773082125678666154970933, 0.921206915201657954834574957219, 0.945066903836358747274923995209, 0.981037062714141615566084749747, 1.32448698846827370000132225603, 1.45202324460079378358063871590, 1.70814922313784356994346333417, 1.71405545678666072810966990178, 1.82784581193390095900187224554, 2.03612154319808083085572565231, 2.19164304065141788144265792194, 2.23985211080503827597293540726, 2.44298193785771087230921645348, 2.49741506707946857156691976439, 2.57542467649560812565451861393, 2.71802474960887502227126960641, 2.73415429501887836121790070821, 2.76298966005525945584519974841, 2.80768075655858865802505780177
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.