Dirichlet series
| L(s) = 1 | + 5·3-s + 4-s + 34·7-s + 21·9-s + 5·12-s + 2·13-s + 34·16-s − 66·19-s + 170·21-s − 138·25-s + 45·27-s + 34·28-s − 126·31-s + 21·36-s − 230·37-s + 10·39-s + 156·43-s + 170·48-s + 761·49-s + 2·52-s − 330·57-s − 104·61-s + 714·63-s + 71·64-s + 368·67-s − 354·73-s − 690·75-s + ⋯ |
| L(s) = 1 | + 5/3·3-s + 1/4·4-s + 34/7·7-s + 7/3·9-s + 5/12·12-s + 2/13·13-s + 17/8·16-s − 3.47·19-s + 8.09·21-s − 5.51·25-s + 5/3·27-s + 1.21·28-s − 4.06·31-s + 7/12·36-s − 6.21·37-s + 0.256·39-s + 3.62·43-s + 3.54·48-s + 15.5·49-s + 1/26·52-s − 5.78·57-s − 1.70·61-s + 34/3·63-s + 1.10·64-s + 5.49·67-s − 4.84·73-s − 9.19·75-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
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.001032970687\) |
| \(L(\frac12)\) | \(\approx\) | \(0.001032970687\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - 5 T + 4 T^{2} + 40 T^{3} - 230 T^{4} + 205 p T^{5} - 17 p^{3} T^{6} - 190 p^{3} T^{7} + 344 p^{4} T^{8} - 190 p^{5} T^{9} - 17 p^{7} T^{10} + 205 p^{7} T^{11} - 230 p^{8} T^{12} + 40 p^{10} T^{13} + 4 p^{12} T^{14} - 5 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - T^{2} - 33 T^{4} - p^{2} T^{6} + 181 T^{8} + 95 p^{4} T^{10} + 4149 p T^{12} - 9535 p T^{14} - 208103 T^{16} - 9535 p^{5} T^{18} + 4149 p^{9} T^{20} + 95 p^{16} T^{22} + 181 p^{16} T^{24} - p^{22} T^{26} - 33 p^{24} T^{28} - p^{28} T^{30} + p^{32} T^{32} \) |
| 5 | \( 1 + 138 T^{2} + 1444 p T^{4} + 90738 T^{6} - 8109291 T^{8} - 434135958 T^{10} - 48111664 p^{3} T^{12} + 203275758492 T^{14} + 10077367518881 T^{16} + 203275758492 p^{4} T^{18} - 48111664 p^{11} T^{20} - 434135958 p^{12} T^{22} - 8109291 p^{16} T^{24} + 90738 p^{20} T^{26} + 1444 p^{25} T^{28} + 138 p^{28} T^{30} + p^{32} T^{32} \) | |
| 7 | \( ( 1 - 17 T + 53 T^{2} + 94 p T^{3} - 5004 T^{4} + 8065 T^{5} - 26868 T^{6} - 219270 T^{7} + 6153877 T^{8} - 219270 p^{2} T^{9} - 26868 p^{4} T^{10} + 8065 p^{6} T^{11} - 5004 p^{8} T^{12} + 94 p^{11} T^{13} + 53 p^{12} T^{14} - 17 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( ( 1 - T - 209 T^{2} + 1750 T^{3} + 278 T^{4} - 894055 T^{5} + 662438 p T^{6} + 90259084 T^{7} - 1572458507 T^{8} + 90259084 p^{2} T^{9} + 662438 p^{5} T^{10} - 894055 p^{6} T^{11} + 278 p^{8} T^{12} + 1750 p^{10} T^{13} - 209 p^{12} T^{14} - p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 + 684 T^{2} + 17875 p T^{4} + 117322710 T^{6} + 27979965225 T^{8} + 2262329617242 T^{10} - 1385214439907207 T^{12} - 978027878460947100 T^{14} - \)\(35\!\cdots\!20\)\( T^{16} - 978027878460947100 p^{4} T^{18} - 1385214439907207 p^{8} T^{20} + 2262329617242 p^{12} T^{22} + 27979965225 p^{16} T^{24} + 117322710 p^{20} T^{26} + 17875 p^{25} T^{28} + 684 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 + 33 T + 338 T^{2} + 14484 T^{3} + 537702 T^{4} + 9315675 T^{5} + 199865947 T^{6} + 238360512 p T^{7} + 226673240 p^{2} T^{8} + 238360512 p^{3} T^{9} + 199865947 p^{4} T^{10} + 9315675 p^{6} T^{11} + 537702 p^{8} T^{12} + 14484 p^{10} T^{13} + 338 p^{12} T^{14} + 33 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 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 | \( 1 + 3432 T^{2} + 6895298 T^{4} + 10498406856 T^{6} + 13894403539107 T^{8} + 16121137948915080 T^{10} + 16888876473633331492 T^{12} + \)\(16\!\cdots\!32\)\( T^{14} + \)\(14\!\cdots\!05\)\( T^{16} + \)\(16\!\cdots\!32\)\( p^{4} T^{18} + 16888876473633331492 p^{8} T^{20} + 16121137948915080 p^{12} T^{22} + 13894403539107 p^{16} T^{24} + 10498406856 p^{20} T^{26} + 6895298 p^{24} T^{28} + 3432 p^{28} T^{30} + p^{32} T^{32} \) | |
| 31 | \( ( 1 + 63 T - 364 T^{2} - 76368 T^{3} - 138534 T^{4} + 98516979 T^{5} + 4008736069 T^{6} - 12937643394 T^{7} - 4346982206488 T^{8} - 12937643394 p^{2} T^{9} + 4008736069 p^{4} T^{10} + 98516979 p^{6} T^{11} - 138534 p^{8} T^{12} - 76368 p^{10} T^{13} - 364 p^{12} T^{14} + 63 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 115 T + 5065 T^{2} + 82310 T^{3} + 296894 T^{4} + 113418535 T^{5} + 4977320390 T^{6} - 163713260410 T^{7} - 15093585180719 T^{8} - 163713260410 p^{2} T^{9} + 4977320390 p^{4} T^{10} + 113418535 p^{6} T^{11} + 296894 p^{8} T^{12} + 82310 p^{10} T^{13} + 5065 p^{12} T^{14} + 115 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( 1 + 5010 T^{2} + 7667066 T^{4} - 4820337435 T^{6} - 20318964751095 T^{8} + 26711744368445760 T^{10} + \)\(10\!\cdots\!49\)\( T^{12} - \)\(92\!\cdots\!75\)\( T^{14} - \)\(28\!\cdots\!06\)\( T^{16} - \)\(92\!\cdots\!75\)\( p^{4} T^{18} + \)\(10\!\cdots\!49\)\( p^{8} T^{20} + 26711744368445760 p^{12} T^{22} - 20318964751095 p^{16} T^{24} - 4820337435 p^{20} T^{26} + 7667066 p^{24} T^{28} + 5010 p^{28} T^{30} + p^{32} T^{32} \) | |
| 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 | \( 1 + 7430 T^{2} + 17869896 T^{4} - 6217911910 T^{6} - 102761006929415 T^{8} - 94532126239899010 T^{10} + \)\(29\!\cdots\!24\)\( T^{12} + \)\(55\!\cdots\!20\)\( T^{14} + \)\(28\!\cdots\!29\)\( T^{16} + \)\(55\!\cdots\!20\)\( p^{4} T^{18} + \)\(29\!\cdots\!24\)\( p^{8} T^{20} - 94532126239899010 p^{12} T^{22} - 102761006929415 p^{16} T^{24} - 6217911910 p^{20} T^{26} + 17869896 p^{24} T^{28} + 7430 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 + 6977 T^{2} + 33417546 T^{4} + 127071731468 T^{6} + 325207594645246 T^{8} + 840512454129903491 T^{10} + \)\(20\!\cdots\!39\)\( T^{12} + \)\(10\!\cdots\!28\)\( p T^{14} + \)\(17\!\cdots\!12\)\( T^{16} + \)\(10\!\cdots\!28\)\( p^{5} T^{18} + \)\(20\!\cdots\!39\)\( p^{8} T^{20} + 840512454129903491 p^{12} T^{22} + 325207594645246 p^{16} T^{24} + 127071731468 p^{20} T^{26} + 33417546 p^{24} T^{28} + 6977 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( 1 + 6171 T^{2} + 44595422 T^{4} + 210841518264 T^{6} + 983258408574126 T^{8} + 4411836914538021225 T^{10} + \)\(17\!\cdots\!83\)\( T^{12} + \)\(71\!\cdots\!60\)\( T^{14} + \)\(25\!\cdots\!92\)\( T^{16} + \)\(71\!\cdots\!60\)\( p^{4} T^{18} + \)\(17\!\cdots\!83\)\( p^{8} T^{20} + 4411836914538021225 p^{12} T^{22} + 983258408574126 p^{16} T^{24} + 210841518264 p^{20} T^{26} + 44595422 p^{24} T^{28} + 6171 p^{28} T^{30} + p^{32} T^{32} \) | |
| 61 | \( ( 1 + 52 T + 176 T^{2} - 6782 T^{3} + 21272241 T^{4} - 119458424 T^{5} - 96888832416 T^{6} - 4139301626796 T^{7} + 102078004281937 T^{8} - 4139301626796 p^{2} T^{9} - 96888832416 p^{4} T^{10} - 119458424 p^{6} T^{11} + 21272241 p^{8} T^{12} - 6782 p^{10} T^{13} + 176 p^{12} T^{14} + 52 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 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 | \( 1 + 18118 T^{2} + 203764684 T^{4} + 1804047769250 T^{6} + 13475089516207613 T^{8} + 89065491850900470430 T^{10} + \)\(53\!\cdots\!56\)\( T^{12} + \)\(29\!\cdots\!72\)\( T^{14} + \)\(15\!\cdots\!93\)\( T^{16} + \)\(29\!\cdots\!72\)\( p^{4} T^{18} + \)\(53\!\cdots\!56\)\( p^{8} T^{20} + 89065491850900470430 p^{12} T^{22} + 13475089516207613 p^{16} T^{24} + 1804047769250 p^{20} T^{26} + 203764684 p^{24} T^{28} + 18118 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 + 177 T + 4082 T^{2} - 1261452 T^{3} - 121484448 T^{4} - 6764307015 T^{5} - 226341803507 T^{6} + 44428763016894 T^{7} + 6559632706012520 T^{8} + 44428763016894 p^{2} T^{9} - 226341803507 p^{4} T^{10} - 6764307015 p^{6} T^{11} - 121484448 p^{8} T^{12} - 1261452 p^{10} T^{13} + 4082 p^{12} T^{14} + 177 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 283 T + 31473 T^{2} + 1808854 T^{3} + 64560502 T^{4} - 3197276465 T^{5} - 1164164477298 T^{6} - 118981266655642 T^{7} - 8733080478953975 T^{8} - 118981266655642 p^{2} T^{9} - 1164164477298 p^{4} T^{10} - 3197276465 p^{6} T^{11} + 64560502 p^{8} T^{12} + 1808854 p^{10} T^{13} + 31473 p^{12} T^{14} + 283 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( 1 + 22345 T^{2} + 347862343 T^{4} + 4481366758400 T^{6} + 48397839984929408 T^{8} + \)\(46\!\cdots\!45\)\( T^{10} + \)\(39\!\cdots\!06\)\( T^{12} + \)\(31\!\cdots\!50\)\( T^{14} + \)\(22\!\cdots\!05\)\( T^{16} + \)\(31\!\cdots\!50\)\( p^{4} T^{18} + \)\(39\!\cdots\!06\)\( p^{8} T^{20} + \)\(46\!\cdots\!45\)\( p^{12} T^{22} + 48397839984929408 p^{16} T^{24} + 4481366758400 p^{20} T^{26} + 347862343 p^{24} T^{28} + 22345 p^{28} T^{30} + p^{32} T^{32} \) | |
| 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 | \( ( 1 - 126 T - 7528 T^{2} + 1771938 T^{3} + 31347897 T^{4} - 1011693150 T^{5} - 1856285365352 T^{6} - 39075032501604 T^{7} + 30063085371156665 T^{8} - 39075032501604 p^{2} T^{9} - 1856285365352 p^{4} T^{10} - 1011693150 p^{6} T^{11} + 31347897 p^{8} T^{12} + 1771938 p^{10} T^{13} - 7528 p^{12} T^{14} - 126 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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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
−2.98255112159218567279175695399, −2.50303742478640828903222311717, −2.49679039755020517234846591576, −2.46708706201758019800616779911, −2.36867298976533343821066579844, −2.31867445843659785534572971597, −2.27495594540916611182646647733, −2.22753699456408722657177997450, −2.17323962917187874092954162784, −2.02111132144867730103260959104, −1.98209486308606667053659842421, −1.83956152026507587209111052364, −1.80608820250242311271402594827, −1.74851663234610380880341802017, −1.52558812653928729788333337509, −1.39939643275922483795621172930, −1.37846956967135771874599908094, −1.31926158916856479645758259274, −1.18772127531013355395984510220, −1.10118123998169449971017074445, −0.858615631274671755417776633188, −0.71911170830890336559998061193, −0.25685624416305938509149528046, −0.15276416005495285600017210460, −0.00219611808759956971861782557, 0.00219611808759956971861782557, 0.15276416005495285600017210460, 0.25685624416305938509149528046, 0.71911170830890336559998061193, 0.858615631274671755417776633188, 1.10118123998169449971017074445, 1.18772127531013355395984510220, 1.31926158916856479645758259274, 1.37846956967135771874599908094, 1.39939643275922483795621172930, 1.52558812653928729788333337509, 1.74851663234610380880341802017, 1.80608820250242311271402594827, 1.83956152026507587209111052364, 1.98209486308606667053659842421, 2.02111132144867730103260959104, 2.17323962917187874092954162784, 2.22753699456408722657177997450, 2.27495594540916611182646647733, 2.31867445843659785534572971597, 2.36867298976533343821066579844, 2.46708706201758019800616779911, 2.49679039755020517234846591576, 2.50303742478640828903222311717, 2.98255112159218567279175695399
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.