Dirichlet series
| L(s) = 1 | + 16·4-s + 80·7-s + 9·9-s + 96·16-s − 342·19-s + 403·25-s + 1.28e3·28-s + 804·31-s + 144·36-s − 962·37-s + 1.73e3·43-s + 3.61e3·49-s − 4.62e3·61-s + 720·63-s − 706·67-s + 3.29e3·73-s − 5.47e3·76-s − 2.65e3·79-s + 72·81-s + 6.44e3·100-s − 6.37e3·103-s + 754·109-s + 7.68e3·112-s − 5.25e3·121-s + 1.28e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s + 4.31·7-s + 1/3·9-s + 3/2·16-s − 4.12·19-s + 3.22·25-s + 8.63·28-s + 4.65·31-s + 2/3·36-s − 4.27·37-s + 6.14·43-s + 10.5·49-s − 9.69·61-s + 1.43·63-s − 1.28·67-s + 5.28·73-s − 8.25·76-s − 3.78·79-s + 8/81·81-s + 6.44·100-s − 6.10·103-s + 0.662·109-s + 6.47·112-s − 3.94·121-s + 9.31·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.02230\times 10^{6}\) |
| Root analytic conductor: | \(1.57419\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(12.33637601\) |
| \(L(\frac12)\) | \(\approx\) | \(12.33637601\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \) |
| 3 | \( 1 - p^{2} T^{2} + p^{2} T^{4} - 56 p^{4} T^{5} + 658 p^{3} T^{6} + 112 p^{4} T^{7} - 5462 p^{4} T^{8} + 112 p^{7} T^{9} + 658 p^{9} T^{10} - 56 p^{13} T^{11} + p^{14} T^{12} - p^{20} T^{14} + p^{24} T^{16} \) | |
| 7 | \( ( 1 - 40 T + 85 p T^{2} - 2440 p T^{3} + 10168 p^{2} T^{4} - 2440 p^{4} T^{5} + 85 p^{7} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| good | 5 | \( 1 - 403 T^{2} + 80508 T^{4} - 11588123 T^{6} + 1123823297 T^{8} - 22213566336 T^{10} - 16629085192034 T^{12} + 846353268915118 p T^{14} - 641674862552716344 T^{16} + 846353268915118 p^{7} T^{18} - 16629085192034 p^{12} T^{20} - 22213566336 p^{18} T^{22} + 1123823297 p^{24} T^{24} - 11588123 p^{30} T^{26} + 80508 p^{36} T^{28} - 403 p^{42} T^{30} + p^{48} T^{32} \) |
| 11 | \( 1 + 5257 T^{2} + 14636052 T^{4} + 26439308609 T^{6} + 30586806197825 T^{8} + 12197776931270784 T^{10} - 42149657439654411554 T^{12} - \)\(12\!\cdots\!74\)\( T^{14} - \)\(21\!\cdots\!96\)\( T^{16} - \)\(12\!\cdots\!74\)\( p^{6} T^{18} - 42149657439654411554 p^{12} T^{20} + 12197776931270784 p^{18} T^{22} + 30586806197825 p^{24} T^{24} + 26439308609 p^{30} T^{26} + 14636052 p^{36} T^{28} + 5257 p^{42} T^{30} + p^{48} T^{32} \) | |
| 13 | \( ( 1 - 13025 T^{2} + 78138478 T^{4} - 289959829895 T^{6} + 749432012155618 T^{8} - 289959829895 p^{6} T^{10} + 78138478 p^{12} T^{12} - 13025 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 17 | \( 1 - 16522 T^{2} + 137136387 T^{4} - 454575947726 T^{6} - 1321922280689143 T^{8} + 21698870810810206452 T^{10} - \)\(44\!\cdots\!90\)\( p T^{12} - \)\(13\!\cdots\!84\)\( T^{14} + \)\(21\!\cdots\!34\)\( T^{16} - \)\(13\!\cdots\!84\)\( p^{6} T^{18} - \)\(44\!\cdots\!90\)\( p^{13} T^{20} + 21698870810810206452 p^{18} T^{22} - 1321922280689143 p^{24} T^{24} - 454575947726 p^{30} T^{26} + 137136387 p^{36} T^{28} - 16522 p^{42} T^{30} + p^{48} T^{32} \) | |
| 19 | \( ( 1 + 9 p T + 25594 T^{2} + 142623 p T^{3} + 240736861 T^{4} + 19120765608 T^{5} + 1272032534050 T^{6} + 97402340768454 T^{7} + 7360758335905852 T^{8} + 97402340768454 p^{3} T^{9} + 1272032534050 p^{6} T^{10} + 19120765608 p^{9} T^{11} + 240736861 p^{12} T^{12} + 142623 p^{16} T^{13} + 25594 p^{18} T^{14} + 9 p^{22} T^{15} + p^{24} T^{16} )^{2} \) | |
| 23 | \( 1 + 52534 T^{2} + 1310752899 T^{4} + 21428880822482 T^{6} + 279188933237825513 T^{8} + \)\(33\!\cdots\!76\)\( T^{10} + \)\(42\!\cdots\!62\)\( T^{12} + \)\(60\!\cdots\!08\)\( T^{14} + \)\(80\!\cdots\!38\)\( T^{16} + \)\(60\!\cdots\!08\)\( p^{6} T^{18} + \)\(42\!\cdots\!62\)\( p^{12} T^{20} + \)\(33\!\cdots\!76\)\( p^{18} T^{22} + 279188933237825513 p^{24} T^{24} + 21428880822482 p^{30} T^{26} + 1310752899 p^{36} T^{28} + 52534 p^{42} T^{30} + p^{48} T^{32} \) | |
| 29 | \( ( 1 - 122383 T^{2} + 7521462478 T^{4} - 300174759894841 T^{6} + 8557303069605841090 T^{8} - 300174759894841 p^{6} T^{10} + 7521462478 p^{12} T^{12} - 122383 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 402 T + 160900 T^{2} - 43026864 T^{3} + 10665461143 T^{4} - 2032521218820 T^{5} + 397781267218288 T^{6} - 64768359125009094 T^{7} + 11704650836981267560 T^{8} - 64768359125009094 p^{3} T^{9} + 397781267218288 p^{6} T^{10} - 2032521218820 p^{9} T^{11} + 10665461143 p^{12} T^{12} - 43026864 p^{15} T^{13} + 160900 p^{18} T^{14} - 402 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 13 p T + 112380 T^{2} + 1460789 p T^{3} + 15506464805 T^{4} + 62279573136 p T^{5} + 825964835350090 T^{6} + 5049060339097090 p T^{7} + 21576403040344504200 T^{8} + 5049060339097090 p^{4} T^{9} + 825964835350090 p^{6} T^{10} + 62279573136 p^{10} T^{11} + 15506464805 p^{12} T^{12} + 1460789 p^{16} T^{13} + 112380 p^{18} T^{14} + 13 p^{22} T^{15} + p^{24} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 118960 T^{2} + 6259839580 T^{4} - 19973031809456 T^{6} - 13756016580223913978 T^{8} - 19973031809456 p^{6} T^{10} + 6259839580 p^{12} T^{12} + 118960 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 433 T + 232300 T^{2} - 65813497 T^{3} + 25262383030 T^{4} - 65813497 p^{3} T^{5} + 232300 p^{6} T^{6} - 433 p^{9} T^{7} + p^{12} T^{8} )^{4} \) | |
| 47 | \( 1 - 490606 T^{2} + 120540379107 T^{4} - 20279964476710226 T^{6} + \)\(26\!\cdots\!29\)\( T^{8} - \)\(28\!\cdots\!76\)\( T^{10} + \)\(23\!\cdots\!06\)\( T^{12} - \)\(17\!\cdots\!32\)\( T^{14} + \)\(14\!\cdots\!66\)\( T^{16} - \)\(17\!\cdots\!32\)\( p^{6} T^{18} + \)\(23\!\cdots\!06\)\( p^{12} T^{20} - \)\(28\!\cdots\!76\)\( p^{18} T^{22} + \)\(26\!\cdots\!29\)\( p^{24} T^{24} - 20279964476710226 p^{30} T^{26} + 120540379107 p^{36} T^{28} - 490606 p^{42} T^{30} + p^{48} T^{32} \) | |
| 53 | \( 1 + 621217 T^{2} + 162286761384 T^{4} + 32889047160459173 T^{6} + \)\(78\!\cdots\!49\)\( T^{8} + \)\(15\!\cdots\!88\)\( T^{10} + \)\(23\!\cdots\!18\)\( T^{12} + \)\(39\!\cdots\!02\)\( T^{14} + \)\(67\!\cdots\!68\)\( T^{16} + \)\(39\!\cdots\!02\)\( p^{6} T^{18} + \)\(23\!\cdots\!18\)\( p^{12} T^{20} + \)\(15\!\cdots\!88\)\( p^{18} T^{22} + \)\(78\!\cdots\!49\)\( p^{24} T^{24} + 32889047160459173 p^{30} T^{26} + 162286761384 p^{36} T^{28} + 621217 p^{42} T^{30} + p^{48} T^{32} \) | |
| 59 | \( 1 - 613987 T^{2} + 180726782592 T^{4} - 7409410440137327 T^{6} - \)\(99\!\cdots\!95\)\( T^{8} + \)\(34\!\cdots\!60\)\( T^{10} - \)\(26\!\cdots\!26\)\( T^{12} - \)\(10\!\cdots\!54\)\( T^{14} + \)\(39\!\cdots\!16\)\( T^{16} - \)\(10\!\cdots\!54\)\( p^{6} T^{18} - \)\(26\!\cdots\!26\)\( p^{12} T^{20} + \)\(34\!\cdots\!60\)\( p^{18} T^{22} - \)\(99\!\cdots\!95\)\( p^{24} T^{24} - 7409410440137327 p^{30} T^{26} + 180726782592 p^{36} T^{28} - 613987 p^{42} T^{30} + p^{48} T^{32} \) | |
| 61 | \( ( 1 + 2310 T + 3278539 T^{2} + 3464628090 T^{3} + 3013154596153 T^{4} + 2225918245393836 T^{5} + 1433461787125846702 T^{6} + \)\(81\!\cdots\!36\)\( T^{7} + \)\(41\!\cdots\!90\)\( T^{8} + \)\(81\!\cdots\!36\)\( p^{3} T^{9} + 1433461787125846702 p^{6} T^{10} + 2225918245393836 p^{9} T^{11} + 3013154596153 p^{12} T^{12} + 3464628090 p^{15} T^{13} + 3278539 p^{18} T^{14} + 2310 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 353 T - 942900 T^{2} - 244510715 T^{3} + 566948576999 T^{4} + 98232161087148 T^{5} - 237551254877958032 T^{6} - 11972800207589977936 T^{7} + \)\(80\!\cdots\!24\)\( T^{8} - 11972800207589977936 p^{3} T^{9} - 237551254877958032 p^{6} T^{10} + 98232161087148 p^{9} T^{11} + 566948576999 p^{12} T^{12} - 244510715 p^{15} T^{13} - 942900 p^{18} T^{14} + 353 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 2185120 T^{2} + 2150648987644 T^{4} - 1293185431312772704 T^{6} + \)\(54\!\cdots\!14\)\( T^{8} - 1293185431312772704 p^{6} T^{10} + 2150648987644 p^{12} T^{12} - 2185120 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 1647 T + 2578552 T^{2} - 2757652803 T^{3} + 2801731240765 T^{4} - 2310432135763680 T^{5} + 1841570048813171362 T^{6} - \)\(12\!\cdots\!62\)\( T^{7} + \)\(84\!\cdots\!28\)\( T^{8} - \)\(12\!\cdots\!62\)\( p^{3} T^{9} + 1841570048813171362 p^{6} T^{10} - 2310432135763680 p^{9} T^{11} + 2801731240765 p^{12} T^{12} - 2757652803 p^{15} T^{13} + 2578552 p^{18} T^{14} - 1647 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 1328 T + 649554 T^{2} + 531584272 T^{3} + 343179705869 T^{4} + 48459420350472 T^{5} + 156782033512738618 T^{6} + \)\(12\!\cdots\!48\)\( T^{7} + \)\(23\!\cdots\!08\)\( T^{8} + \)\(12\!\cdots\!48\)\( p^{3} T^{9} + 156782033512738618 p^{6} T^{10} + 48459420350472 p^{9} T^{11} + 343179705869 p^{12} T^{12} + 531584272 p^{15} T^{13} + 649554 p^{18} T^{14} + 1328 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 1722229 T^{2} + 1384053320998 T^{4} + 728536700729120515 T^{6} + \)\(37\!\cdots\!94\)\( T^{8} + 728536700729120515 p^{6} T^{10} + 1384053320998 p^{12} T^{12} + 1722229 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 89 | \( 1 - 2323450 T^{2} + 1452170603619 T^{4} - 832430879304658430 T^{6} + \)\(23\!\cdots\!61\)\( T^{8} - \)\(21\!\cdots\!20\)\( T^{10} + \)\(62\!\cdots\!02\)\( T^{12} - \)\(86\!\cdots\!80\)\( T^{14} + \)\(11\!\cdots\!78\)\( T^{16} - \)\(86\!\cdots\!80\)\( p^{6} T^{18} + \)\(62\!\cdots\!02\)\( p^{12} T^{20} - \)\(21\!\cdots\!20\)\( p^{18} T^{22} + \)\(23\!\cdots\!61\)\( p^{24} T^{24} - 832430879304658430 p^{30} T^{26} + 1452170603619 p^{36} T^{28} - 2323450 p^{42} T^{30} + p^{48} T^{32} \) | |
| 97 | \( ( 1 - 6468653 T^{2} + 18847452442330 T^{4} - 32473627146088049315 T^{6} + \)\(36\!\cdots\!06\)\( T^{8} - 32473627146088049315 p^{6} T^{10} + 18847452442330 p^{12} T^{12} - 6468653 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.57056450433916717597063304265, −4.53216331203246032270720255213, −4.45882152661551908755427190304, −4.35828057829564904324543467280, −4.28256869631074807568177319796, −4.27778485419507690254107983749, −4.11717668506248095010404118767, −3.96520433960802959689091070994, −3.55601823750063915409284878848, −3.49329648580107071537752523870, −3.28149325960991510892352155582, −2.91819872575656085997738859166, −2.88586236306103118510840540375, −2.84514711539733920434908448416, −2.58873764215124798777500597205, −2.49783835648320092326696177864, −2.45680303265975874847664943562, −2.00242272473797244360681682431, −1.98082379158613252076274990060, −1.85197233113115655136756724682, −1.50511377620774824331762936665, −1.23784660790389368779327579723, −1.21442106286669964666587283670, −1.10176397680589148327476835046, −0.28038133432348573849515631207, 0.28038133432348573849515631207, 1.10176397680589148327476835046, 1.21442106286669964666587283670, 1.23784660790389368779327579723, 1.50511377620774824331762936665, 1.85197233113115655136756724682, 1.98082379158613252076274990060, 2.00242272473797244360681682431, 2.45680303265975874847664943562, 2.49783835648320092326696177864, 2.58873764215124798777500597205, 2.84514711539733920434908448416, 2.88586236306103118510840540375, 2.91819872575656085997738859166, 3.28149325960991510892352155582, 3.49329648580107071537752523870, 3.55601823750063915409284878848, 3.96520433960802959689091070994, 4.11717668506248095010404118767, 4.27778485419507690254107983749, 4.28256869631074807568177319796, 4.35828057829564904324543467280, 4.45882152661551908755427190304, 4.53216331203246032270720255213, 4.57056450433916717597063304265
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.