Dirichlet series
| L(s) = 1 | − 40·9-s + 24·13-s − 4·23-s − 40·25-s + 32·27-s − 108·29-s + 116·31-s − 156·41-s + 128·47-s + 378·49-s − 204·59-s − 236·71-s − 112·73-s + 622·81-s + 100·101-s − 960·117-s + 920·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 476·169-s + ⋯ |
| L(s) = 1 | − 4.44·9-s + 1.84·13-s − 0.173·23-s − 8/5·25-s + 1.18·27-s − 3.72·29-s + 3.74·31-s − 3.80·41-s + 2.72·47-s + 54/7·49-s − 3.45·59-s − 3.32·71-s − 1.53·73-s + 7.67·81-s + 0.990·101-s − 8.20·117-s + 7.60·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.81·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 5^{16} \cdot 23^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.59384\times 10^{27}\) |
| Root analytic conductor: | \(7.08070\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 5^{16} \cdot 23^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.007403287686\) |
| \(L(\frac12)\) | \(\approx\) | \(0.007403287686\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p T^{2} )^{8} \) | |
| 23 | \( 1 + 4 T - 638 T^{2} + 26988 T^{3} + 13672 p T^{4} - 31708 p^{2} T^{5} + 36562 p^{3} T^{6} + 27180 p^{4} T^{7} - 44254 p^{5} T^{8} + 27180 p^{6} T^{9} + 36562 p^{7} T^{10} - 31708 p^{8} T^{11} + 13672 p^{9} T^{12} + 26988 p^{10} T^{13} - 638 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} \) | |
| good | 3 | \( ( 1 + 20 T^{2} - 16 T^{3} + 289 T^{4} - 88 p T^{5} + 3374 T^{6} - 4352 T^{7} + 31288 T^{8} - 4352 p^{2} T^{9} + 3374 p^{4} T^{10} - 88 p^{7} T^{11} + 289 p^{8} T^{12} - 16 p^{10} T^{13} + 20 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 7 | \( 1 - 54 p T^{2} + 77505 T^{4} - 1584138 p T^{6} + 1218487602 T^{8} - 108111071602 T^{10} + 7966903359055 T^{12} - 10117756624686 p^{2} T^{14} + 10946984829818 p^{4} T^{16} - 10117756624686 p^{6} T^{18} + 7966903359055 p^{8} T^{20} - 108111071602 p^{12} T^{22} + 1218487602 p^{16} T^{24} - 1584138 p^{21} T^{26} + 77505 p^{24} T^{28} - 54 p^{29} T^{30} + p^{32} T^{32} \) | |
| 11 | \( 1 - 920 T^{2} + 471792 T^{4} - 15306488 p T^{6} + 46169132060 T^{8} - 10195318729976 T^{10} + 1866454544289552 T^{12} - 287802276947650504 T^{14} + 37725410953602366790 T^{16} - 287802276947650504 p^{4} T^{18} + 1866454544289552 p^{8} T^{20} - 10195318729976 p^{12} T^{22} + 46169132060 p^{16} T^{24} - 15306488 p^{21} T^{26} + 471792 p^{24} T^{28} - 920 p^{28} T^{30} + p^{32} T^{32} \) | |
| 13 | \( ( 1 - 12 T + 454 T^{2} - 9084 T^{3} + 188769 T^{4} - 2665120 T^{5} + 53038074 T^{6} - 678500168 T^{7} + 9551823100 T^{8} - 678500168 p^{2} T^{9} + 53038074 p^{4} T^{10} - 2665120 p^{6} T^{11} + 188769 p^{8} T^{12} - 9084 p^{10} T^{13} + 454 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 - 2766 T^{2} + 3778037 T^{4} - 3385544366 T^{6} + 131584665238 p T^{8} - 1163374563695522 T^{10} + 496905279815081211 T^{12} - \)\(17\!\cdots\!42\)\( T^{14} + \)\(55\!\cdots\!98\)\( T^{16} - \)\(17\!\cdots\!42\)\( p^{4} T^{18} + 496905279815081211 p^{8} T^{20} - 1163374563695522 p^{12} T^{22} + 131584665238 p^{17} T^{24} - 3385544366 p^{20} T^{26} + 3778037 p^{24} T^{28} - 2766 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( 1 - 1592 T^{2} + 1511920 T^{4} - 1009452936 T^{6} + 521225711388 T^{8} - 217136126963608 T^{10} + 77189263445867536 T^{12} - 25288522323204185512 T^{14} + \)\(86\!\cdots\!90\)\( T^{16} - 25288522323204185512 p^{4} T^{18} + 77189263445867536 p^{8} T^{20} - 217136126963608 p^{12} T^{22} + 521225711388 p^{16} T^{24} - 1009452936 p^{20} T^{26} + 1511920 p^{24} T^{28} - 1592 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( ( 1 + 54 T + 4419 T^{2} + 164522 T^{3} + 8308921 T^{4} + 262001836 T^{5} + 10635715550 T^{6} + 301091017096 T^{7} + 10330577872738 T^{8} + 301091017096 p^{2} T^{9} + 10635715550 p^{4} T^{10} + 262001836 p^{6} T^{11} + 8308921 p^{8} T^{12} + 164522 p^{10} T^{13} + 4419 p^{12} T^{14} + 54 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 58 T + 165 p T^{2} - 248786 T^{3} + 13490457 T^{4} - 543362992 T^{5} + 22344370250 T^{6} - 757700682124 T^{7} + 826825140618 p T^{8} - 757700682124 p^{2} T^{9} + 22344370250 p^{4} T^{10} - 543362992 p^{6} T^{11} + 13490457 p^{8} T^{12} - 248786 p^{10} T^{13} + 165 p^{13} T^{14} - 58 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( 1 - 7422 T^{2} + 31726061 T^{4} - 99536214046 T^{6} + 251265395645742 T^{8} - 536179744077194162 T^{10} + \)\(99\!\cdots\!75\)\( T^{12} - \)\(16\!\cdots\!90\)\( T^{14} + \)\(23\!\cdots\!86\)\( T^{16} - \)\(16\!\cdots\!90\)\( p^{4} T^{18} + \)\(99\!\cdots\!75\)\( p^{8} T^{20} - 536179744077194162 p^{12} T^{22} + 251265395645742 p^{16} T^{24} - 99536214046 p^{20} T^{26} + 31726061 p^{24} T^{28} - 7422 p^{28} T^{30} + p^{32} T^{32} \) | |
| 41 | \( ( 1 + 78 T + 9287 T^{2} + 11894 p T^{3} + 32155605 T^{4} + 1224054504 T^{5} + 59192080134 T^{6} + 1858208531132 T^{7} + 89322055594606 T^{8} + 1858208531132 p^{2} T^{9} + 59192080134 p^{4} T^{10} + 1224054504 p^{6} T^{11} + 32155605 p^{8} T^{12} + 11894 p^{11} T^{13} + 9287 p^{12} T^{14} + 78 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 43 | \( 1 - 16172 T^{2} + 125748812 T^{4} - 635198769972 T^{6} + 2388920603563444 T^{8} - 7274642887007437148 T^{10} + \)\(18\!\cdots\!08\)\( T^{12} - \)\(42\!\cdots\!48\)\( T^{14} + \)\(84\!\cdots\!14\)\( T^{16} - \)\(42\!\cdots\!48\)\( p^{4} T^{18} + \)\(18\!\cdots\!08\)\( p^{8} T^{20} - 7274642887007437148 p^{12} T^{22} + 2388920603563444 p^{16} T^{24} - 635198769972 p^{20} T^{26} + 125748812 p^{24} T^{28} - 16172 p^{28} T^{30} + p^{32} T^{32} \) | |
| 47 | \( ( 1 - 64 T + 11908 T^{2} - 479608 T^{3} + 52771465 T^{4} - 23229736 p T^{5} + 119742536086 T^{6} - 382130197304 T^{7} + 223115145998696 T^{8} - 382130197304 p^{2} T^{9} + 119742536086 p^{4} T^{10} - 23229736 p^{7} T^{11} + 52771465 p^{8} T^{12} - 479608 p^{10} T^{13} + 11908 p^{12} T^{14} - 64 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 53 | \( 1 - 23694 T^{2} + 257544965 T^{4} - 1673958546206 T^{6} + 7047260182331126 T^{8} - 18821435882716772594 T^{10} + \)\(23\!\cdots\!35\)\( T^{12} + \)\(35\!\cdots\!14\)\( T^{14} - \)\(23\!\cdots\!90\)\( T^{16} + \)\(35\!\cdots\!14\)\( p^{4} T^{18} + \)\(23\!\cdots\!35\)\( p^{8} T^{20} - 18821435882716772594 p^{12} T^{22} + 7047260182331126 p^{16} T^{24} - 1673958546206 p^{20} T^{26} + 257544965 p^{24} T^{28} - 23694 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( ( 1 + 102 T + 13385 T^{2} + 802586 T^{3} + 75582938 T^{4} + 3551043038 T^{5} + 301501022071 T^{6} + 12063023997122 T^{7} + 1059908106439210 T^{8} + 12063023997122 p^{2} T^{9} + 301501022071 p^{4} T^{10} + 3551043038 p^{6} T^{11} + 75582938 p^{8} T^{12} + 802586 p^{10} T^{13} + 13385 p^{12} T^{14} + 102 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 61 | \( 1 - 31408 T^{2} + 502811408 T^{4} - 5424340278704 T^{6} + 44159901024954076 T^{8} - \)\(28\!\cdots\!56\)\( T^{10} + \)\(15\!\cdots\!56\)\( T^{12} - \)\(72\!\cdots\!00\)\( T^{14} + \)\(28\!\cdots\!18\)\( T^{16} - \)\(72\!\cdots\!00\)\( p^{4} T^{18} + \)\(15\!\cdots\!56\)\( p^{8} T^{20} - \)\(28\!\cdots\!56\)\( p^{12} T^{22} + 44159901024954076 p^{16} T^{24} - 5424340278704 p^{20} T^{26} + 502811408 p^{24} T^{28} - 31408 p^{28} T^{30} + p^{32} T^{32} \) | |
| 67 | \( 1 - 19146 T^{2} + 195649009 T^{4} - 1467789804278 T^{6} + 9451325273979506 T^{8} - 55830598070664400674 T^{10} + \)\(30\!\cdots\!15\)\( T^{12} - \)\(15\!\cdots\!94\)\( T^{14} + \)\(72\!\cdots\!78\)\( T^{16} - \)\(15\!\cdots\!94\)\( p^{4} T^{18} + \)\(30\!\cdots\!15\)\( p^{8} T^{20} - 55830598070664400674 p^{12} T^{22} + 9451325273979506 p^{16} T^{24} - 1467789804278 p^{20} T^{26} + 195649009 p^{24} T^{28} - 19146 p^{28} T^{30} + p^{32} T^{32} \) | |
| 71 | \( ( 1 + 118 T + 29195 T^{2} + 2738974 T^{3} + 396005833 T^{4} + 31734907232 T^{5} + 3407028116186 T^{6} + 235070254631348 T^{7} + 20387187187755350 T^{8} + 235070254631348 p^{2} T^{9} + 3407028116186 p^{4} T^{10} + 31734907232 p^{6} T^{11} + 396005833 p^{8} T^{12} + 2738974 p^{10} T^{13} + 29195 p^{12} T^{14} + 118 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 56 T + 27818 T^{2} + 1805112 T^{3} + 380646465 T^{4} + 341672896 p T^{5} + 3411931086186 T^{6} + 201904374476976 T^{7} + 21553009649391316 T^{8} + 201904374476976 p^{2} T^{9} + 3411931086186 p^{4} T^{10} + 341672896 p^{7} T^{11} + 380646465 p^{8} T^{12} + 1805112 p^{10} T^{13} + 27818 p^{12} T^{14} + 56 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( 1 - 17640 T^{2} + 154954256 T^{4} - 1049025531160 T^{6} + 7172033075875804 T^{8} - 51017335955237366600 T^{10} + \)\(31\!\cdots\!56\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{14} + \)\(86\!\cdots\!70\)\( T^{16} - \)\(16\!\cdots\!20\)\( p^{4} T^{18} + \)\(31\!\cdots\!56\)\( p^{8} T^{20} - 51017335955237366600 p^{12} T^{22} + 7172033075875804 p^{16} T^{24} - 1049025531160 p^{20} T^{26} + 154954256 p^{24} T^{28} - 17640 p^{28} T^{30} + p^{32} T^{32} \) | |
| 83 | \( 1 - 40362 T^{2} + 944805737 T^{4} - 15728699881302 T^{6} + 207953675196789914 T^{8} - \)\(22\!\cdots\!58\)\( T^{10} + \)\(21\!\cdots\!03\)\( T^{12} - \)\(17\!\cdots\!98\)\( T^{14} + \)\(13\!\cdots\!54\)\( T^{16} - \)\(17\!\cdots\!98\)\( p^{4} T^{18} + \)\(21\!\cdots\!03\)\( p^{8} T^{20} - \)\(22\!\cdots\!58\)\( p^{12} T^{22} + 207953675196789914 p^{16} T^{24} - 15728699881302 p^{20} T^{26} + 944805737 p^{24} T^{28} - 40362 p^{28} T^{30} + p^{32} T^{32} \) | |
| 89 | \( 1 - 58808 T^{2} + 1790517168 T^{4} - 37850217863816 T^{6} + 621590980604295452 T^{8} - \)\(83\!\cdots\!32\)\( T^{10} + \)\(95\!\cdots\!64\)\( T^{12} - \)\(93\!\cdots\!60\)\( T^{14} + \)\(79\!\cdots\!62\)\( T^{16} - \)\(93\!\cdots\!60\)\( p^{4} T^{18} + \)\(95\!\cdots\!64\)\( p^{8} T^{20} - \)\(83\!\cdots\!32\)\( p^{12} T^{22} + 621590980604295452 p^{16} T^{24} - 37850217863816 p^{20} T^{26} + 1790517168 p^{24} T^{28} - 58808 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( 1 - 66688 T^{2} + 2319385584 T^{4} - 56457830279200 T^{6} + 1077137668741357852 T^{8} - \)\(16\!\cdots\!56\)\( T^{10} + \)\(22\!\cdots\!72\)\( T^{12} - \)\(26\!\cdots\!68\)\( T^{14} + \)\(26\!\cdots\!82\)\( T^{16} - \)\(26\!\cdots\!68\)\( p^{4} T^{18} + \)\(22\!\cdots\!72\)\( p^{8} T^{20} - \)\(16\!\cdots\!56\)\( p^{12} T^{22} + 1077137668741357852 p^{16} T^{24} - 56457830279200 p^{20} T^{26} + 2319385584 p^{24} T^{28} - 66688 p^{28} T^{30} + p^{32} T^{32} \) | |
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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
−2.21746557675891319668995927615, −1.85706742117281920621236092465, −1.83039672072311190754237782987, −1.81153596413117049310131877464, −1.79902133912593976954955757530, −1.77482099324541576795625107471, −1.76237847281188859745311873383, −1.66748380117080390098607543823, −1.46755162026363560439918106665, −1.29767451796897737776188232003, −1.25902073406475322939345089143, −1.23998092803239903190513761429, −1.23852660578252320534378091338, −1.16129295393907946161641365386, −0.957496746062487071995766917168, −0.823381990938221509340697357126, −0.77536027375349841428228559625, −0.65999323953931098404716605330, −0.56167408804772714229406656280, −0.54810949288055566130484655123, −0.45985987390455745902790483605, −0.39863439576490086645484120514, −0.18605769196419663344829685327, −0.06213670297023895847672683568, −0.01113508585788618775112025097, 0.01113508585788618775112025097, 0.06213670297023895847672683568, 0.18605769196419663344829685327, 0.39863439576490086645484120514, 0.45985987390455745902790483605, 0.54810949288055566130484655123, 0.56167408804772714229406656280, 0.65999323953931098404716605330, 0.77536027375349841428228559625, 0.823381990938221509340697357126, 0.957496746062487071995766917168, 1.16129295393907946161641365386, 1.23852660578252320534378091338, 1.23998092803239903190513761429, 1.25902073406475322939345089143, 1.29767451796897737776188232003, 1.46755162026363560439918106665, 1.66748380117080390098607543823, 1.76237847281188859745311873383, 1.77482099324541576795625107471, 1.79902133912593976954955757530, 1.81153596413117049310131877464, 1.83039672072311190754237782987, 1.85706742117281920621236092465, 2.21746557675891319668995927615
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.