Dirichlet series
| L(s) = 1 | − 18·5-s + 26·7-s − 720·11-s + 10·13-s − 100·19-s + 1.27e3·23-s − 691·25-s + 1.85e3·29-s + 1.47e3·31-s − 468·35-s − 32·37-s + 36·41-s + 68·43-s + 2.21e3·47-s + 6.36e3·49-s + 1.29e4·55-s + 9.10e3·59-s − 4.47e3·61-s − 180·65-s − 7.50e3·67-s + 2.07e4·73-s − 1.87e4·77-s + 6.05e3·79-s − 3.83e3·83-s + 260·91-s + 1.80e3·95-s + 3.13e4·97-s + ⋯ |
| L(s) = 1 | − 0.719·5-s + 0.530·7-s − 5.95·11-s + 0.0591·13-s − 0.277·19-s + 2.41·23-s − 1.10·25-s + 2.20·29-s + 1.53·31-s − 0.382·35-s − 0.0233·37-s + 0.0214·41-s + 0.0367·43-s + 1.00·47-s + 2.64·49-s + 4.28·55-s + 2.61·59-s − 1.20·61-s − 0.0426·65-s − 1.67·67-s + 3.88·73-s − 3.15·77-s + 0.969·79-s − 0.556·83-s + 0.0313·91-s + 0.199·95-s + 3.33·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.58133\times 10^{13}\) |
| Root analytic conductor: | \(6.68250\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{24} ,\ ( \ : [2]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(3.350883959\) |
| \(L(\frac12)\) | \(\approx\) | \(3.350883959\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 18 T + 203 p T^{2} + 16326 T^{3} + 22357 p^{2} T^{4} + 1081188 p T^{5} - 314464142 T^{6} - 702270504 p T^{7} - 306662326214 T^{8} - 702270504 p^{5} T^{9} - 314464142 p^{8} T^{10} + 1081188 p^{13} T^{11} + 22357 p^{18} T^{12} + 16326 p^{20} T^{13} + 203 p^{25} T^{14} + 18 p^{28} T^{15} + p^{32} T^{16} \) |
| 7 | \( 1 - 26 T - 5685 T^{2} - 19510 T^{3} + 19835513 T^{4} + 334899084 T^{5} - 45037679042 T^{6} - 451520935736 T^{7} + 90278690161986 T^{8} - 451520935736 p^{4} T^{9} - 45037679042 p^{8} T^{10} + 334899084 p^{12} T^{11} + 19835513 p^{16} T^{12} - 19510 p^{20} T^{13} - 5685 p^{24} T^{14} - 26 p^{28} T^{15} + p^{32} T^{16} \) | |
| 11 | \( 1 + 720 T + 24962 p T^{2} + 73283040 T^{3} + 15179596969 T^{4} + 2592718024176 T^{5} + 383108185780198 T^{6} + 51080491982360160 T^{7} + 6354846385614434932 T^{8} + 51080491982360160 p^{4} T^{9} + 383108185780198 p^{8} T^{10} + 2592718024176 p^{12} T^{11} + 15179596969 p^{16} T^{12} + 73283040 p^{20} T^{13} + 24962 p^{25} T^{14} + 720 p^{28} T^{15} + p^{32} T^{16} \) | |
| 13 | \( 1 - 10 T - 34545 T^{2} - 8013734 T^{3} + 152683301 T^{4} + 264027759132 T^{5} + 34691703939994 T^{6} - 3939272844487792 T^{7} - 880372532161082934 T^{8} - 3939272844487792 p^{4} T^{9} + 34691703939994 p^{8} T^{10} + 264027759132 p^{12} T^{11} + 152683301 p^{16} T^{12} - 8013734 p^{20} T^{13} - 34545 p^{24} T^{14} - 10 p^{28} T^{15} + p^{32} T^{16} \) | |
| 17 | \( 1 - 249218 T^{2} + 30482460289 T^{4} - 3595961342423810 T^{6} + \)\(36\!\cdots\!56\)\( T^{8} - 3595961342423810 p^{8} T^{10} + 30482460289 p^{16} T^{12} - 249218 p^{24} T^{14} + p^{32} T^{16} \) | |
| 19 | \( ( 1 + 50 T + 226153 T^{2} + 12428354 T^{3} + 33132078964 T^{4} + 12428354 p^{4} T^{5} + 226153 p^{8} T^{6} + 50 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 23 | \( 1 - 1278 T + 1655467 T^{2} - 1419907842 T^{3} + 1186049018761 T^{4} - 817434220578732 T^{5} + 550327618738318606 T^{6} - \)\(32\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!30\)\( T^{8} - \)\(32\!\cdots\!56\)\( p^{4} T^{9} + 550327618738318606 p^{8} T^{10} - 817434220578732 p^{12} T^{11} + 1186049018761 p^{16} T^{12} - 1419907842 p^{20} T^{13} + 1655467 p^{24} T^{14} - 1278 p^{28} T^{15} + p^{32} T^{16} \) | |
| 29 | \( 1 - 1854 T + 3116071 T^{2} - 3652934346 T^{3} + 3878113467181 T^{4} - 4083633101435148 T^{5} + 4044828167734324498 T^{6} - \)\(39\!\cdots\!28\)\( T^{7} + \)\(34\!\cdots\!30\)\( T^{8} - \)\(39\!\cdots\!28\)\( p^{4} T^{9} + 4044828167734324498 p^{8} T^{10} - 4083633101435148 p^{12} T^{11} + 3878113467181 p^{16} T^{12} - 3652934346 p^{20} T^{13} + 3116071 p^{24} T^{14} - 1854 p^{28} T^{15} + p^{32} T^{16} \) | |
| 31 | \( 1 - 1478 T - 273609 T^{2} + 14717462 T^{3} + 1239820588133 T^{4} + 59691395608740 T^{5} - 148444619433025670 T^{6} + \)\(17\!\cdots\!40\)\( T^{7} - \)\(10\!\cdots\!22\)\( T^{8} + \)\(17\!\cdots\!40\)\( p^{4} T^{9} - 148444619433025670 p^{8} T^{10} + 59691395608740 p^{12} T^{11} + 1239820588133 p^{16} T^{12} + 14717462 p^{20} T^{13} - 273609 p^{24} T^{14} - 1478 p^{28} T^{15} + p^{32} T^{16} \) | |
| 37 | \( ( 1 + 16 T + 5152060 T^{2} + 1816976752 T^{3} + 11981316770374 T^{4} + 1816976752 p^{4} T^{5} + 5152060 p^{8} T^{6} + 16 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 41 | \( 1 - 36 T + 10981366 T^{2} - 395313624 T^{3} + 74491912981249 T^{4} - 2021455678408920 T^{5} + \)\(33\!\cdots\!50\)\( T^{6} - \)\(74\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!68\)\( T^{8} - \)\(74\!\cdots\!60\)\( p^{4} T^{9} + \)\(33\!\cdots\!50\)\( p^{8} T^{10} - 2021455678408920 p^{12} T^{11} + 74491912981249 p^{16} T^{12} - 395313624 p^{20} T^{13} + 10981366 p^{24} T^{14} - 36 p^{28} T^{15} + p^{32} T^{16} \) | |
| 43 | \( 1 - 68 T - 249966 p T^{2} - 2055805384 T^{3} + 65375958867329 T^{4} + 16226764390392264 T^{5} - \)\(28\!\cdots\!82\)\( T^{6} - \)\(26\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!76\)\( T^{8} - \)\(26\!\cdots\!32\)\( p^{4} T^{9} - \)\(28\!\cdots\!82\)\( p^{8} T^{10} + 16226764390392264 p^{12} T^{11} + 65375958867329 p^{16} T^{12} - 2055805384 p^{20} T^{13} - 249966 p^{25} T^{14} - 68 p^{28} T^{15} + p^{32} T^{16} \) | |
| 47 | \( 1 - 2214 T + 16322779 T^{2} - 32521107258 T^{3} + 136095840713065 T^{4} - 267869759907619980 T^{5} + \)\(91\!\cdots\!74\)\( T^{6} - \)\(16\!\cdots\!08\)\( T^{7} + \)\(50\!\cdots\!62\)\( T^{8} - \)\(16\!\cdots\!08\)\( p^{4} T^{9} + \)\(91\!\cdots\!74\)\( p^{8} T^{10} - 267869759907619980 p^{12} T^{11} + 136095840713065 p^{16} T^{12} - 32521107258 p^{20} T^{13} + 16322779 p^{24} T^{14} - 2214 p^{28} T^{15} + p^{32} T^{16} \) | |
| 53 | \( 1 - 47149064 T^{2} + 1030375014657436 T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!10\)\( T^{8} - \)\(13\!\cdots\!20\)\( p^{8} T^{10} + 1030375014657436 p^{16} T^{12} - 47149064 p^{24} T^{14} + p^{32} T^{16} \) | |
| 59 | \( 1 - 9108 T + 65431078 T^{2} - 344092862520 T^{3} + 1463362600081057 T^{4} - 4890955357569963960 T^{5} + \)\(14\!\cdots\!50\)\( T^{6} - \)\(36\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!92\)\( T^{8} - \)\(36\!\cdots\!52\)\( p^{4} T^{9} + \)\(14\!\cdots\!50\)\( p^{8} T^{10} - 4890955357569963960 p^{12} T^{11} + 1463362600081057 p^{16} T^{12} - 344092862520 p^{20} T^{13} + 65431078 p^{24} T^{14} - 9108 p^{28} T^{15} + p^{32} T^{16} \) | |
| 61 | \( 1 + 4478 T - 125937 T^{2} - 125289669758 T^{3} - 568176190027387 T^{4} - 742254823045377828 T^{5} + \)\(27\!\cdots\!38\)\( T^{6} + \)\(22\!\cdots\!92\)\( T^{7} + \)\(73\!\cdots\!38\)\( T^{8} + \)\(22\!\cdots\!92\)\( p^{4} T^{9} + \)\(27\!\cdots\!38\)\( p^{8} T^{10} - 742254823045377828 p^{12} T^{11} - 568176190027387 p^{16} T^{12} - 125289669758 p^{20} T^{13} - 125937 p^{24} T^{14} + 4478 p^{28} T^{15} + p^{32} T^{16} \) | |
| 67 | \( 1 + 112 p T - 22347594 T^{2} - 226068030400 T^{3} + 738296226968777 T^{4} + 4521766862440091376 T^{5} - \)\(18\!\cdots\!90\)\( T^{6} - \)\(18\!\cdots\!56\)\( T^{7} + \)\(57\!\cdots\!20\)\( T^{8} - \)\(18\!\cdots\!56\)\( p^{4} T^{9} - \)\(18\!\cdots\!90\)\( p^{8} T^{10} + 4521766862440091376 p^{12} T^{11} + 738296226968777 p^{16} T^{12} - 226068030400 p^{20} T^{13} - 22347594 p^{24} T^{14} + 112 p^{29} T^{15} + p^{32} T^{16} \) | |
| 71 | \( 1 - 125290160 T^{2} + 8106610376011420 T^{4} - \)\(34\!\cdots\!88\)\( T^{6} + \)\(10\!\cdots\!18\)\( T^{8} - \)\(34\!\cdots\!88\)\( p^{8} T^{10} + 8106610376011420 p^{16} T^{12} - 125290160 p^{24} T^{14} + p^{32} T^{16} \) | |
| 73 | \( ( 1 - 10358 T + 106803217 T^{2} - 769838088062 T^{3} + 4439859073965124 T^{4} - 769838088062 p^{4} T^{5} + 106803217 p^{8} T^{6} - 10358 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 79 | \( 1 - 6050 T - 42146805 T^{2} - 267987207070 T^{3} + 3321952223354537 T^{4} + 14945323617952326060 T^{5} + \)\(48\!\cdots\!70\)\( T^{6} - \)\(67\!\cdots\!20\)\( T^{7} - \)\(27\!\cdots\!02\)\( T^{8} - \)\(67\!\cdots\!20\)\( p^{4} T^{9} + \)\(48\!\cdots\!70\)\( p^{8} T^{10} + 14945323617952326060 p^{12} T^{11} + 3321952223354537 p^{16} T^{12} - 267987207070 p^{20} T^{13} - 42146805 p^{24} T^{14} - 6050 p^{28} T^{15} + p^{32} T^{16} \) | |
| 83 | \( 1 + 3834 T + 163867483 T^{2} + 609481897254 T^{3} + 15130998622630921 T^{4} + 49888358310056197140 T^{5} + \)\(10\!\cdots\!30\)\( T^{6} + \)\(30\!\cdots\!80\)\( T^{7} + \)\(54\!\cdots\!10\)\( T^{8} + \)\(30\!\cdots\!80\)\( p^{4} T^{9} + \)\(10\!\cdots\!30\)\( p^{8} T^{10} + 49888358310056197140 p^{12} T^{11} + 15130998622630921 p^{16} T^{12} + 609481897254 p^{20} T^{13} + 163867483 p^{24} T^{14} + 3834 p^{28} T^{15} + p^{32} T^{16} \) | |
| 89 | \( 1 - 206196488 T^{2} + 19314405141094684 T^{4} - \)\(14\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!86\)\( T^{8} - \)\(14\!\cdots\!60\)\( p^{8} T^{10} + 19314405141094684 p^{16} T^{12} - 206196488 p^{24} T^{14} + p^{32} T^{16} \) | |
| 97 | \( 1 - 31336 T + 307771782 T^{2} - 2149219106864 T^{3} + 52690258168920329 T^{4} - \)\(68\!\cdots\!16\)\( T^{5} + \)\(38\!\cdots\!30\)\( T^{6} - \)\(46\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!04\)\( T^{8} - \)\(46\!\cdots\!00\)\( p^{4} T^{9} + \)\(38\!\cdots\!30\)\( p^{8} T^{10} - \)\(68\!\cdots\!16\)\( p^{12} T^{11} + 52690258168920329 p^{16} T^{12} - 2149219106864 p^{20} T^{13} + 307771782 p^{24} T^{14} - 31336 p^{28} T^{15} + p^{32} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.31900623604604928670995845856, −4.07108165914936706942131031982, −4.01594692327522238962915785750, −3.64744540983650355161303258821, −3.56058490362246539716809214486, −3.55314013230219870015730708548, −3.23584033991297930761234258841, −3.17455504941373954531557125434, −2.88741240394112960018488378280, −2.76075538499665955264909230938, −2.72573374289041714869288481067, −2.51908719828407893755603568336, −2.46954298670904439954922883766, −2.31819170197406519140560050988, −2.11002346581938572013556743942, −1.96542275136986925693656522306, −1.91564673148899839464080773448, −1.33558020594744939463217346383, −1.33541653759423142729385930930, −0.819638902402317198238245933672, −0.73288150931695760635245618510, −0.68944338112703560003894843694, −0.56302008044745273083974461456, −0.45350124555757276179529430198, −0.12836437118565978563415168936, 0.12836437118565978563415168936, 0.45350124555757276179529430198, 0.56302008044745273083974461456, 0.68944338112703560003894843694, 0.73288150931695760635245618510, 0.819638902402317198238245933672, 1.33541653759423142729385930930, 1.33558020594744939463217346383, 1.91564673148899839464080773448, 1.96542275136986925693656522306, 2.11002346581938572013556743942, 2.31819170197406519140560050988, 2.46954298670904439954922883766, 2.51908719828407893755603568336, 2.72573374289041714869288481067, 2.76075538499665955264909230938, 2.88741240394112960018488378280, 3.17455504941373954531557125434, 3.23584033991297930761234258841, 3.55314013230219870015730708548, 3.56058490362246539716809214486, 3.64744540983650355161303258821, 4.01594692327522238962915785750, 4.07108165914936706942131031982, 4.31900623604604928670995845856
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.