Dirichlet series
| L(s) = 1 | − 32·2-s + 384·4-s − 528·5-s − 560·7-s + 1.68e4·10-s − 2.16e3·11-s − 1.34e4·13-s + 1.79e4·14-s − 6.14e4·16-s + 4.51e4·17-s + 7.34e4·19-s − 2.02e5·20-s + 6.91e4·22-s − 6.26e4·23-s + 2.48e5·25-s + 4.30e5·26-s − 2.15e5·28-s − 6.84e4·29-s − 2.27e5·31-s + 7.86e5·32-s − 1.44e6·34-s + 2.95e5·35-s + 1.04e6·37-s − 2.34e6·38-s − 6.72e4·41-s − 5.62e5·43-s − 8.29e5·44-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s − 1.88·5-s − 0.617·7-s + 5.34·10-s − 0.489·11-s − 1.69·13-s + 1.74·14-s − 3.75·16-s + 2.22·17-s + 2.45·19-s − 5.66·20-s + 1.38·22-s − 1.07·23-s + 3.17·25-s + 4.80·26-s − 1.85·28-s − 0.520·29-s − 1.37·31-s + 4.24·32-s − 6.30·34-s + 1.16·35-s + 3.39·37-s − 6.94·38-s − 0.152·41-s − 1.07·43-s − 1.46·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.30171\times 10^{13}\) |
| Root analytic conductor: | \(7.11381\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [7/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(1.039010534\) |
| \(L(\frac12)\) | \(\approx\) | \(1.039010534\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{3} T + p^{6} T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 528 T + 6098 p T^{2} - 28717728 T^{3} - 6646473479 T^{4} - 296584780896 p T^{5} - 2892551971246 p^{3} T^{6} + 302393999457264 p^{3} T^{7} + 55894546867697956 p^{4} T^{8} + 302393999457264 p^{10} T^{9} - 2892551971246 p^{17} T^{10} - 296584780896 p^{22} T^{11} - 6646473479 p^{28} T^{12} - 28717728 p^{35} T^{13} + 6098 p^{43} T^{14} + 528 p^{49} T^{15} + p^{56} T^{16} \) |
| 7 | \( 1 + 80 p T - 765588 T^{2} + 4467040 p^{2} T^{3} + 383373171338 T^{4} - 7906783464240 p^{2} T^{5} + 685455739329691696 T^{6} + 72837084001705034960 p T^{7} - \)\(47\!\cdots\!21\)\( T^{8} + 72837084001705034960 p^{8} T^{9} + 685455739329691696 p^{14} T^{10} - 7906783464240 p^{23} T^{11} + 383373171338 p^{28} T^{12} + 4467040 p^{37} T^{13} - 765588 p^{42} T^{14} + 80 p^{50} T^{15} + p^{56} T^{16} \) | |
| 11 | \( 1 + 2160 T - 34124660 T^{2} - 31418508960 T^{3} + 360882118725466 T^{4} - 1046783223117817200 T^{5} - \)\(91\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!80\)\( T^{7} + \)\(32\!\cdots\!75\)\( T^{8} + \)\(19\!\cdots\!80\)\( p^{7} T^{9} - \)\(91\!\cdots\!20\)\( p^{14} T^{10} - 1046783223117817200 p^{21} T^{11} + 360882118725466 p^{28} T^{12} - 31418508960 p^{35} T^{13} - 34124660 p^{42} T^{14} + 2160 p^{49} T^{15} + p^{56} T^{16} \) | |
| 13 | \( 1 + 13460 T - 2381874 T^{2} - 1006670971640 T^{3} - 486455445387667 p T^{4} - 19004114515218025800 T^{5} - \)\(49\!\cdots\!06\)\( T^{6} + \)\(18\!\cdots\!80\)\( T^{7} + \)\(27\!\cdots\!24\)\( T^{8} + \)\(18\!\cdots\!80\)\( p^{7} T^{9} - \)\(49\!\cdots\!06\)\( p^{14} T^{10} - 19004114515218025800 p^{21} T^{11} - 486455445387667 p^{29} T^{12} - 1006670971640 p^{35} T^{13} - 2381874 p^{42} T^{14} + 13460 p^{49} T^{15} + p^{56} T^{16} \) | |
| 17 | \( ( 1 - 22560 T + 652500158 T^{2} - 256565445120 T^{3} + 29387435907228099 T^{4} - 256565445120 p^{7} T^{5} + 652500158 p^{14} T^{6} - 22560 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 19 | \( ( 1 - 36704 T + 3047986756 T^{2} - 91971206881760 T^{3} + 3917212049965000630 T^{4} - 91971206881760 p^{7} T^{5} + 3047986756 p^{14} T^{6} - 36704 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 23 | \( 1 + 62640 T - 285060028 p T^{2} - 64514052819360 T^{3} + 43613736591780304234 T^{4} - \)\(46\!\cdots\!00\)\( T^{5} - \)\(16\!\cdots\!96\)\( T^{6} + \)\(37\!\cdots\!40\)\( p T^{7} + \)\(68\!\cdots\!31\)\( p^{2} T^{8} + \)\(37\!\cdots\!40\)\( p^{8} T^{9} - \)\(16\!\cdots\!96\)\( p^{14} T^{10} - \)\(46\!\cdots\!00\)\( p^{21} T^{11} + 43613736591780304234 p^{28} T^{12} - 64514052819360 p^{35} T^{13} - 285060028 p^{43} T^{14} + 62640 p^{49} T^{15} + p^{56} T^{16} \) | |
| 29 | \( 1 + 68400 T - 22912702982 T^{2} + 3119287342989600 T^{3} + \)\(41\!\cdots\!29\)\( T^{4} - \)\(81\!\cdots\!00\)\( T^{5} + \)\(67\!\cdots\!94\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} - \)\(12\!\cdots\!80\)\( T^{8} + \)\(12\!\cdots\!00\)\( p^{7} T^{9} + \)\(67\!\cdots\!94\)\( p^{14} T^{10} - \)\(81\!\cdots\!00\)\( p^{21} T^{11} + \)\(41\!\cdots\!29\)\( p^{28} T^{12} + 3119287342989600 p^{35} T^{13} - 22912702982 p^{42} T^{14} + 68400 p^{49} T^{15} + p^{56} T^{16} \) | |
| 31 | \( 1 + 227504 T - 21465016380 T^{2} + 4801891406924128 T^{3} + \)\(23\!\cdots\!58\)\( T^{4} - \)\(16\!\cdots\!72\)\( T^{5} - \)\(14\!\cdots\!32\)\( T^{6} + \)\(34\!\cdots\!40\)\( T^{7} - \)\(62\!\cdots\!93\)\( T^{8} + \)\(34\!\cdots\!40\)\( p^{7} T^{9} - \)\(14\!\cdots\!32\)\( p^{14} T^{10} - \)\(16\!\cdots\!72\)\( p^{21} T^{11} + \)\(23\!\cdots\!58\)\( p^{28} T^{12} + 4801891406924128 p^{35} T^{13} - 21465016380 p^{42} T^{14} + 227504 p^{49} T^{15} + p^{56} T^{16} \) | |
| 37 | \( ( 1 - 523580 T + 359904908818 T^{2} - 143318985284154800 T^{3} + \)\(13\!\cdots\!83\)\( p T^{4} - 143318985284154800 p^{7} T^{5} + 359904908818 p^{14} T^{6} - 523580 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 41 | \( 1 + 67200 T - 515575757828 T^{2} - 39340714365093120 T^{3} + \)\(14\!\cdots\!54\)\( T^{4} + \)\(94\!\cdots\!40\)\( T^{5} - \)\(25\!\cdots\!24\)\( T^{6} - \)\(97\!\cdots\!60\)\( T^{7} + \)\(43\!\cdots\!55\)\( T^{8} - \)\(97\!\cdots\!60\)\( p^{7} T^{9} - \)\(25\!\cdots\!24\)\( p^{14} T^{10} + \)\(94\!\cdots\!40\)\( p^{21} T^{11} + \)\(14\!\cdots\!54\)\( p^{28} T^{12} - 39340714365093120 p^{35} T^{13} - 515575757828 p^{42} T^{14} + 67200 p^{49} T^{15} + p^{56} T^{16} \) | |
| 43 | \( 1 + 562640 T - 420776333892 T^{2} - 138346669673028320 T^{3} + \)\(15\!\cdots\!78\)\( T^{4} + \)\(16\!\cdots\!20\)\( T^{5} - \)\(29\!\cdots\!96\)\( T^{6} - \)\(43\!\cdots\!60\)\( T^{7} + \)\(82\!\cdots\!59\)\( T^{8} - \)\(43\!\cdots\!60\)\( p^{7} T^{9} - \)\(29\!\cdots\!96\)\( p^{14} T^{10} + \)\(16\!\cdots\!20\)\( p^{21} T^{11} + \)\(15\!\cdots\!78\)\( p^{28} T^{12} - 138346669673028320 p^{35} T^{13} - 420776333892 p^{42} T^{14} + 562640 p^{49} T^{15} + p^{56} T^{16} \) | |
| 47 | \( 1 + 515328 T - 1041981772508 T^{2} - 616082234310551040 T^{3} + \)\(45\!\cdots\!02\)\( T^{4} + \)\(21\!\cdots\!12\)\( T^{5} - \)\(29\!\cdots\!48\)\( T^{6} - \)\(17\!\cdots\!72\)\( T^{7} + \)\(21\!\cdots\!31\)\( T^{8} - \)\(17\!\cdots\!72\)\( p^{7} T^{9} - \)\(29\!\cdots\!48\)\( p^{14} T^{10} + \)\(21\!\cdots\!12\)\( p^{21} T^{11} + \)\(45\!\cdots\!02\)\( p^{28} T^{12} - 616082234310551040 p^{35} T^{13} - 1041981772508 p^{42} T^{14} + 515328 p^{49} T^{15} + p^{56} T^{16} \) | |
| 53 | \( ( 1 - 2498016 T + 4667252414900 T^{2} - 6949193609921122080 T^{3} + \)\(86\!\cdots\!98\)\( T^{4} - 6949193609921122080 p^{7} T^{5} + 4667252414900 p^{14} T^{6} - 2498016 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 59 | \( 1 + 4155840 T + 1386493873972 T^{2} - 3148704644236145280 T^{3} + \)\(41\!\cdots\!18\)\( T^{4} + \)\(74\!\cdots\!20\)\( T^{5} - \)\(36\!\cdots\!32\)\( T^{6} + \)\(71\!\cdots\!80\)\( T^{7} + \)\(50\!\cdots\!11\)\( T^{8} + \)\(71\!\cdots\!80\)\( p^{7} T^{9} - \)\(36\!\cdots\!32\)\( p^{14} T^{10} + \)\(74\!\cdots\!20\)\( p^{21} T^{11} + \)\(41\!\cdots\!18\)\( p^{28} T^{12} - 3148704644236145280 p^{35} T^{13} + 1386493873972 p^{42} T^{14} + 4155840 p^{49} T^{15} + p^{56} T^{16} \) | |
| 61 | \( 1 + 2130764 T - 3014625418578 T^{2} + 2642708038716008248 T^{3} + \)\(20\!\cdots\!45\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{5} - \)\(17\!\cdots\!42\)\( T^{6} + \)\(53\!\cdots\!88\)\( T^{7} - \)\(35\!\cdots\!16\)\( T^{8} + \)\(53\!\cdots\!88\)\( p^{7} T^{9} - \)\(17\!\cdots\!42\)\( p^{14} T^{10} - \)\(21\!\cdots\!40\)\( p^{21} T^{11} + \)\(20\!\cdots\!45\)\( p^{28} T^{12} + 2642708038716008248 p^{35} T^{13} - 3014625418578 p^{42} T^{14} + 2130764 p^{49} T^{15} + p^{56} T^{16} \) | |
| 67 | \( 1 - 1205440 T - 3106269411588 T^{2} + 35326918145455542400 T^{3} - \)\(33\!\cdots\!62\)\( T^{4} - \)\(14\!\cdots\!80\)\( T^{5} + \)\(46\!\cdots\!76\)\( T^{6} + \)\(50\!\cdots\!60\)\( T^{7} - \)\(31\!\cdots\!81\)\( T^{8} + \)\(50\!\cdots\!60\)\( p^{7} T^{9} + \)\(46\!\cdots\!76\)\( p^{14} T^{10} - \)\(14\!\cdots\!80\)\( p^{21} T^{11} - \)\(33\!\cdots\!62\)\( p^{28} T^{12} + 35326918145455542400 p^{35} T^{13} - 3106269411588 p^{42} T^{14} - 1205440 p^{49} T^{15} + p^{56} T^{16} \) | |
| 71 | \( ( 1 - 12486480 T + 89708735416388 T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!98\)\( T^{4} - \)\(43\!\cdots\!00\)\( p^{7} T^{5} + 89708735416388 p^{14} T^{6} - 12486480 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 4820860 T + 38589071904874 T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(60\!\cdots\!87\)\( T^{4} + \)\(15\!\cdots\!40\)\( p^{7} T^{5} + 38589071904874 p^{14} T^{6} + 4820860 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 79 | \( 1 - 11471680 T + 63121310835564 T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} + \)\(41\!\cdots\!40\)\( p T^{5} - \)\(14\!\cdots\!64\)\( T^{6} + \)\(29\!\cdots\!20\)\( T^{7} - \)\(53\!\cdots\!41\)\( T^{8} + \)\(29\!\cdots\!20\)\( p^{7} T^{9} - \)\(14\!\cdots\!64\)\( p^{14} T^{10} + \)\(41\!\cdots\!40\)\( p^{22} T^{11} + \)\(11\!\cdots\!10\)\( p^{28} T^{12} - \)\(20\!\cdots\!60\)\( p^{35} T^{13} + 63121310835564 p^{42} T^{14} - 11471680 p^{49} T^{15} + p^{56} T^{16} \) | |
| 83 | \( 1 + 16811232 T + 150404417986132 T^{2} + \)\(60\!\cdots\!64\)\( T^{3} - \)\(72\!\cdots\!22\)\( T^{4} - \)\(24\!\cdots\!24\)\( T^{5} - \)\(12\!\cdots\!76\)\( T^{6} - \)\(14\!\cdots\!48\)\( T^{7} + \)\(88\!\cdots\!23\)\( T^{8} - \)\(14\!\cdots\!48\)\( p^{7} T^{9} - \)\(12\!\cdots\!76\)\( p^{14} T^{10} - \)\(24\!\cdots\!24\)\( p^{21} T^{11} - \)\(72\!\cdots\!22\)\( p^{28} T^{12} + \)\(60\!\cdots\!64\)\( p^{35} T^{13} + 150404417986132 p^{42} T^{14} + 16811232 p^{49} T^{15} + p^{56} T^{16} \) | |
| 89 | \( ( 1 - 16857600 T + 212149846489550 T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!99\)\( T^{4} - \)\(19\!\cdots\!00\)\( p^{7} T^{5} + 212149846489550 p^{14} T^{6} - 16857600 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 97 | \( 1 - 27078040 T + 195301397964900 T^{2} - \)\(83\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!74\)\( T^{4} - \)\(33\!\cdots\!40\)\( T^{5} + \)\(22\!\cdots\!00\)\( T^{6} - \)\(89\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!15\)\( T^{8} - \)\(89\!\cdots\!20\)\( p^{7} T^{9} + \)\(22\!\cdots\!00\)\( p^{14} T^{10} - \)\(33\!\cdots\!40\)\( p^{21} T^{11} + \)\(31\!\cdots\!74\)\( p^{28} T^{12} - \)\(83\!\cdots\!20\)\( p^{35} T^{13} + 195301397964900 p^{42} T^{14} - 27078040 p^{49} T^{15} + p^{56} T^{16} \) | |
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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
−4.42268968357139324854598370906, −4.35211546290866933134079740307, −3.99303616185649615453702956596, −3.95166585076624984543250409010, −3.75569407739017611735964207410, −3.54367337666225955696133688588, −3.39659163818577323590618674441, −3.36503305745231424375723216326, −3.15281217337257346854871928542, −3.02872539211603706842819759593, −2.59420234343364714792271985969, −2.43832735664410802550404803762, −2.34479546983453816364319512662, −2.34117534416299320622998616371, −1.85422764342332677160315632962, −1.69387552978732328603835821663, −1.66232577799690050456100248789, −1.08305847857385341894271348662, −1.02227770150265118239995720475, −0.880380918656767767778154988007, −0.844046379494394512789461709819, −0.58132281598952325456499859311, −0.55395323705862797919098087384, −0.31853878932003977321339357356, −0.20780042925037711965450096921, 0.20780042925037711965450096921, 0.31853878932003977321339357356, 0.55395323705862797919098087384, 0.58132281598952325456499859311, 0.844046379494394512789461709819, 0.880380918656767767778154988007, 1.02227770150265118239995720475, 1.08305847857385341894271348662, 1.66232577799690050456100248789, 1.69387552978732328603835821663, 1.85422764342332677160315632962, 2.34117534416299320622998616371, 2.34479546983453816364319512662, 2.43832735664410802550404803762, 2.59420234343364714792271985969, 3.02872539211603706842819759593, 3.15281217337257346854871928542, 3.36503305745231424375723216326, 3.39659163818577323590618674441, 3.54367337666225955696133688588, 3.75569407739017611735964207410, 3.95166585076624984543250409010, 3.99303616185649615453702956596, 4.35211546290866933134079740307, 4.42268968357139324854598370906
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.