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 − 2.85·21-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 + 24/7·63-s − 0.218·64-s + 5.49·67-s + 0.328·73-s − 2.26·75-s + 7.05·76-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.182634\) |
| Root analytic conductor: | \(0.948253\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 11^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.001557717064\) |
| \(L(\frac12)\) | \(\approx\) | \(0.001557717064\) |
| \(L(2)\) | 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 | \( 1 - 72 T^{2} - 2867 p T^{4} + 306 p^{3} T^{6} + 375 p^{6} T^{8} + 306 p^{7} T^{10} - 2867 p^{9} T^{12} - 72 p^{12} T^{14} + p^{16} T^{16} \) | |
| 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
−5.46674441313917918928535994250, −5.40887361338099353619873721221, −5.36594415819512618964682143260, −4.89829003821551526995068207709, −4.85907748114585960391054737003, −4.79320695853155673146892708004, −4.79124953153297098143636343104, −4.60058385696875865140689475670, −4.56294848408930895543033444433, −4.53125510935500220645658031454, −4.26144291725557982852523893967, −4.18421250827964618658076547380, −4.06239140290557108042790310609, −3.98771700021863636673754337924, −3.72148618207764170458922966311, −3.47296378048369802836646352750, −3.05440350754701718694872391010, −2.80683410649141799733119501045, −2.79130138692981389927296819600, −2.42736319382287273952130768570, −2.39612955760726099805731489055, −2.33579253706577828701355707008, −1.93702728513667002522623300336, −1.23799141357952671671960465085, −0.05399574569636960424370906896, 0.05399574569636960424370906896, 1.23799141357952671671960465085, 1.93702728513667002522623300336, 2.33579253706577828701355707008, 2.39612955760726099805731489055, 2.42736319382287273952130768570, 2.79130138692981389927296819600, 2.80683410649141799733119501045, 3.05440350754701718694872391010, 3.47296378048369802836646352750, 3.72148618207764170458922966311, 3.98771700021863636673754337924, 4.06239140290557108042790310609, 4.18421250827964618658076547380, 4.26144291725557982852523893967, 4.53125510935500220645658031454, 4.56294848408930895543033444433, 4.60058385696875865140689475670, 4.79124953153297098143636343104, 4.79320695853155673146892708004, 4.85907748114585960391054737003, 4.89829003821551526995068207709, 5.36594415819512618964682143260, 5.40887361338099353619873721221, 5.46674441313917918928535994250
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.