Dirichlet series
| L(s) = 1 | + 6·2-s + 28·3-s + 100·4-s + 252·5-s + 168·6-s + 384·8-s + 3.76e3·9-s + 1.51e3·10-s + 3.97e3·11-s + 2.80e3·12-s + 2.35e3·13-s + 7.05e3·15-s + 848·16-s + 5.63e4·17-s + 2.25e4·18-s + 4.17e4·19-s + 2.52e4·20-s + 2.38e4·22-s − 1.31e5·23-s + 1.07e4·24-s + 2.22e5·25-s + 1.41e4·26-s + 1.28e4·27-s + 6.81e4·29-s + 4.23e4·30-s + 4.01e5·31-s + 5.52e4·32-s + ⋯ |
| L(s) = 1 | + 0.530·2-s + 0.598·3-s + 0.781·4-s + 0.901·5-s + 0.317·6-s + 0.265·8-s + 1.72·9-s + 0.478·10-s + 0.899·11-s + 0.467·12-s + 0.296·13-s + 0.539·15-s + 0.0517·16-s + 2.78·17-s + 0.912·18-s + 1.39·19-s + 0.704·20-s + 0.477·22-s − 2.25·23-s + 0.158·24-s + 2.84·25-s + 0.157·26-s + 0.125·27-s + 0.518·29-s + 0.286·30-s + 2.41·31-s + 0.297·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.01363\times 10^{9}\) |
| Root analytic conductor: | \(3.91239\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 7^{16} ,\ ( \ : [7/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(94.30513705\) |
| \(L(\frac12)\) | \(\approx\) | \(94.30513705\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 3 p T - p^{6} T^{2} + 75 p^{3} T^{3} + 133 p^{5} T^{4} - 3471 p^{5} T^{5} + 46275 p^{6} T^{6} - 32577 p^{8} T^{7} - 315199 p^{10} T^{8} - 32577 p^{15} T^{9} + 46275 p^{20} T^{10} - 3471 p^{26} T^{11} + 133 p^{33} T^{12} + 75 p^{38} T^{13} - p^{48} T^{14} - 3 p^{50} T^{15} + p^{56} T^{16} \) |
| 3 | \( 1 - 28 T - 2978 T^{2} + 19544 p^{2} T^{3} - 75301 p^{3} T^{4} - 5578048 p^{3} T^{5} + 4404170 p^{5} T^{6} - 93832844 p^{7} T^{7} + 44741366116 p^{6} T^{8} - 93832844 p^{14} T^{9} + 4404170 p^{19} T^{10} - 5578048 p^{24} T^{11} - 75301 p^{31} T^{12} + 19544 p^{37} T^{13} - 2978 p^{42} T^{14} - 28 p^{49} T^{15} + p^{56} T^{16} \) | |
| 5 | \( 1 - 252 T - 158506 T^{2} + 4582536 p T^{3} + 674342201 p^{2} T^{4} - 795864888 p^{4} T^{5} - 2408486026458 p^{4} T^{6} + 6152884831212 p^{5} T^{7} + 6846672196907924 p^{6} T^{8} + 6152884831212 p^{12} T^{9} - 2408486026458 p^{18} T^{10} - 795864888 p^{25} T^{11} + 674342201 p^{30} T^{12} + 4582536 p^{36} T^{13} - 158506 p^{42} T^{14} - 252 p^{49} T^{15} + p^{56} T^{16} \) | |
| 11 | \( 1 - 3972 T - 44545426 T^{2} + 16974598392 p T^{3} + 1193279820296585 T^{4} - 4311182308727377824 T^{5} - \)\(21\!\cdots\!46\)\( T^{6} + \)\(39\!\cdots\!00\)\( T^{7} + \)\(37\!\cdots\!60\)\( T^{8} + \)\(39\!\cdots\!00\)\( p^{7} T^{9} - \)\(21\!\cdots\!46\)\( p^{14} T^{10} - 4311182308727377824 p^{21} T^{11} + 1193279820296585 p^{28} T^{12} + 16974598392 p^{36} T^{13} - 44545426 p^{42} T^{14} - 3972 p^{49} T^{15} + p^{56} T^{16} \) | |
| 13 | \( ( 1 - 1176 T + 151551116 T^{2} - 19538457960 p T^{3} + 75518676758118 p^{2} T^{4} - 19538457960 p^{8} T^{5} + 151551116 p^{14} T^{6} - 1176 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 17 | \( 1 - 56364 T + 709129558 T^{2} + 4712531411112 T^{3} + 360653231962445473 T^{4} - \)\(17\!\cdots\!28\)\( T^{5} + \)\(20\!\cdots\!10\)\( T^{6} - \)\(22\!\cdots\!96\)\( T^{7} + \)\(74\!\cdots\!84\)\( T^{8} - \)\(22\!\cdots\!96\)\( p^{7} T^{9} + \)\(20\!\cdots\!10\)\( p^{14} T^{10} - \)\(17\!\cdots\!28\)\( p^{21} T^{11} + 360653231962445473 p^{28} T^{12} + 4712531411112 p^{35} T^{13} + 709129558 p^{42} T^{14} - 56364 p^{49} T^{15} + p^{56} T^{16} \) | |
| 19 | \( 1 - 41748 T - 2001962786 T^{2} + 58487933247240 T^{3} + 4448060922484490329 T^{4} - \)\(73\!\cdots\!44\)\( T^{5} - \)\(55\!\cdots\!70\)\( T^{6} + \)\(17\!\cdots\!88\)\( T^{7} + \)\(62\!\cdots\!72\)\( T^{8} + \)\(17\!\cdots\!88\)\( p^{7} T^{9} - \)\(55\!\cdots\!70\)\( p^{14} T^{10} - \)\(73\!\cdots\!44\)\( p^{21} T^{11} + 4448060922484490329 p^{28} T^{12} + 58487933247240 p^{35} T^{13} - 2001962786 p^{42} T^{14} - 41748 p^{49} T^{15} + p^{56} T^{16} \) | |
| 23 | \( 1 + 131748 T + 1744906414 T^{2} - 95298488634120 T^{3} + 26557457139118316761 T^{4} + \)\(59\!\cdots\!56\)\( T^{5} - \)\(14\!\cdots\!10\)\( T^{6} - \)\(42\!\cdots\!36\)\( T^{7} + \)\(15\!\cdots\!44\)\( T^{8} - \)\(42\!\cdots\!36\)\( p^{7} T^{9} - \)\(14\!\cdots\!10\)\( p^{14} T^{10} + \)\(59\!\cdots\!56\)\( p^{21} T^{11} + 26557457139118316761 p^{28} T^{12} - 95298488634120 p^{35} T^{13} + 1744906414 p^{42} T^{14} + 131748 p^{49} T^{15} + p^{56} T^{16} \) | |
| 29 | \( ( 1 - 34056 T + 37434723340 T^{2} + 965899818999528 T^{3} + \)\(63\!\cdots\!58\)\( T^{4} + 965899818999528 p^{7} T^{5} + 37434723340 p^{14} T^{6} - 34056 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 31 | \( 1 - 401212 T + 20995911038 T^{2} + 7710914411317432 T^{3} + \)\(63\!\cdots\!53\)\( T^{4} - \)\(41\!\cdots\!68\)\( T^{5} + \)\(19\!\cdots\!38\)\( T^{6} - \)\(11\!\cdots\!88\)\( T^{7} + \)\(99\!\cdots\!68\)\( T^{8} - \)\(11\!\cdots\!88\)\( p^{7} T^{9} + \)\(19\!\cdots\!38\)\( p^{14} T^{10} - \)\(41\!\cdots\!68\)\( p^{21} T^{11} + \)\(63\!\cdots\!53\)\( p^{28} T^{12} + 7710914411317432 p^{35} T^{13} + 20995911038 p^{42} T^{14} - 401212 p^{49} T^{15} + p^{56} T^{16} \) | |
| 37 | \( 1 - 5396 T - 363098326522 T^{2} + 1692295529079608 T^{3} + \)\(80\!\cdots\!33\)\( T^{4} - \)\(28\!\cdots\!72\)\( T^{5} - \)\(11\!\cdots\!30\)\( T^{6} + \)\(10\!\cdots\!56\)\( T^{7} + \)\(13\!\cdots\!04\)\( T^{8} + \)\(10\!\cdots\!56\)\( p^{7} T^{9} - \)\(11\!\cdots\!30\)\( p^{14} T^{10} - \)\(28\!\cdots\!72\)\( p^{21} T^{11} + \)\(80\!\cdots\!33\)\( p^{28} T^{12} + 1692295529079608 p^{35} T^{13} - 363098326522 p^{42} T^{14} - 5396 p^{49} T^{15} + p^{56} T^{16} \) | |
| 41 | \( ( 1 + 410424 T + 514131659548 T^{2} + 263929507132312392 T^{3} + \)\(12\!\cdots\!54\)\( T^{4} + 263929507132312392 p^{7} T^{5} + 514131659548 p^{14} T^{6} + 410424 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 46544 T + 620213285356 T^{2} - 12386906008354640 T^{3} + \)\(22\!\cdots\!42\)\( T^{4} - 12386906008354640 p^{7} T^{5} + 620213285356 p^{14} T^{6} - 46544 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 47 | \( 1 - 1470084 T - 61407153218 T^{2} + 676586115609875016 T^{3} + \)\(33\!\cdots\!21\)\( T^{4} - \)\(36\!\cdots\!44\)\( T^{5} - \)\(22\!\cdots\!18\)\( T^{6} + \)\(49\!\cdots\!56\)\( T^{7} + \)\(16\!\cdots\!52\)\( T^{8} + \)\(49\!\cdots\!56\)\( p^{7} T^{9} - \)\(22\!\cdots\!18\)\( p^{14} T^{10} - \)\(36\!\cdots\!44\)\( p^{21} T^{11} + \)\(33\!\cdots\!21\)\( p^{28} T^{12} + 676586115609875016 p^{35} T^{13} - 61407153218 p^{42} T^{14} - 1470084 p^{49} T^{15} + p^{56} T^{16} \) | |
| 53 | \( 1 - 642372 T - 2597575978538 T^{2} + 2359978517441309784 T^{3} + \)\(29\!\cdots\!33\)\( T^{4} - \)\(28\!\cdots\!24\)\( T^{5} - \)\(30\!\cdots\!50\)\( T^{6} + \)\(12\!\cdots\!88\)\( T^{7} + \)\(40\!\cdots\!84\)\( T^{8} + \)\(12\!\cdots\!88\)\( p^{7} T^{9} - \)\(30\!\cdots\!50\)\( p^{14} T^{10} - \)\(28\!\cdots\!24\)\( p^{21} T^{11} + \)\(29\!\cdots\!33\)\( p^{28} T^{12} + 2359978517441309784 p^{35} T^{13} - 2597575978538 p^{42} T^{14} - 642372 p^{49} T^{15} + p^{56} T^{16} \) | |
| 59 | \( 1 - 752220 T - 3986199031874 T^{2} - 1265706973401854760 T^{3} + \)\(80\!\cdots\!61\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!18\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} - \)\(10\!\cdots\!32\)\( T^{8} - \)\(21\!\cdots\!40\)\( p^{7} T^{9} + \)\(13\!\cdots\!18\)\( p^{14} T^{10} + \)\(10\!\cdots\!20\)\( p^{21} T^{11} + \)\(80\!\cdots\!61\)\( p^{28} T^{12} - 1265706973401854760 p^{35} T^{13} - 3986199031874 p^{42} T^{14} - 752220 p^{49} T^{15} + p^{56} T^{16} \) | |
| 61 | \( 1 - 1325772 T - 2658227552762 T^{2} - 8845336598195007288 T^{3} + \)\(19\!\cdots\!13\)\( T^{4} + \)\(17\!\cdots\!12\)\( T^{5} + \)\(85\!\cdots\!78\)\( T^{6} - \)\(13\!\cdots\!68\)\( T^{7} - \)\(18\!\cdots\!92\)\( T^{8} - \)\(13\!\cdots\!68\)\( p^{7} T^{9} + \)\(85\!\cdots\!78\)\( p^{14} T^{10} + \)\(17\!\cdots\!12\)\( p^{21} T^{11} + \)\(19\!\cdots\!13\)\( p^{28} T^{12} - 8845336598195007288 p^{35} T^{13} - 2658227552762 p^{42} T^{14} - 1325772 p^{49} T^{15} + p^{56} T^{16} \) | |
| 67 | \( 1 - 290916 T - 10943334257858 T^{2} - 8785742150796028056 T^{3} + \)\(62\!\cdots\!41\)\( T^{4} + \)\(82\!\cdots\!44\)\( T^{5} + \)\(15\!\cdots\!02\)\( T^{6} - \)\(38\!\cdots\!96\)\( T^{7} - \)\(21\!\cdots\!88\)\( T^{8} - \)\(38\!\cdots\!96\)\( p^{7} T^{9} + \)\(15\!\cdots\!02\)\( p^{14} T^{10} + \)\(82\!\cdots\!44\)\( p^{21} T^{11} + \)\(62\!\cdots\!41\)\( p^{28} T^{12} - 8785742150796028056 p^{35} T^{13} - 10943334257858 p^{42} T^{14} - 290916 p^{49} T^{15} + p^{56} T^{16} \) | |
| 71 | \( ( 1 - 3377760 T + 21455628109916 T^{2} - 39866994142106473440 T^{3} + \)\(20\!\cdots\!26\)\( T^{4} - 39866994142106473440 p^{7} T^{5} + 21455628109916 p^{14} T^{6} - 3377760 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 73 | \( 1 - 6706588 T - 4081288956538 T^{2} + 75839108854686445576 T^{3} + \)\(20\!\cdots\!73\)\( T^{4} - \)\(78\!\cdots\!56\)\( T^{5} - \)\(36\!\cdots\!10\)\( T^{6} - \)\(16\!\cdots\!88\)\( T^{7} + \)\(80\!\cdots\!84\)\( T^{8} - \)\(16\!\cdots\!88\)\( p^{7} T^{9} - \)\(36\!\cdots\!10\)\( p^{14} T^{10} - \)\(78\!\cdots\!56\)\( p^{21} T^{11} + \)\(20\!\cdots\!73\)\( p^{28} T^{12} + 75839108854686445576 p^{35} T^{13} - 4081288956538 p^{42} T^{14} - 6706588 p^{49} T^{15} + p^{56} T^{16} \) | |
| 79 | \( 1 + 3946244 T - 34598478524562 T^{2} + 37240845779266596472 T^{3} + \)\(11\!\cdots\!45\)\( T^{4} - \)\(24\!\cdots\!12\)\( T^{5} - \)\(13\!\cdots\!86\)\( T^{6} + \)\(27\!\cdots\!48\)\( T^{7} + \)\(56\!\cdots\!44\)\( T^{8} + \)\(27\!\cdots\!48\)\( p^{7} T^{9} - \)\(13\!\cdots\!86\)\( p^{14} T^{10} - \)\(24\!\cdots\!12\)\( p^{21} T^{11} + \)\(11\!\cdots\!45\)\( p^{28} T^{12} + 37240845779266596472 p^{35} T^{13} - 34598478524562 p^{42} T^{14} + 3946244 p^{49} T^{15} + p^{56} T^{16} \) | |
| 83 | \( ( 1 + 9542064 T + 123192866313548 T^{2} + \)\(78\!\cdots\!08\)\( T^{3} + \)\(52\!\cdots\!18\)\( T^{4} + \)\(78\!\cdots\!08\)\( p^{7} T^{5} + 123192866313548 p^{14} T^{6} + 9542064 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 89 | \( 1 - 16165212 T + 27735348124934 T^{2} + \)\(31\!\cdots\!60\)\( T^{3} + \)\(50\!\cdots\!09\)\( T^{4} - \)\(34\!\cdots\!96\)\( T^{5} - \)\(22\!\cdots\!10\)\( T^{6} - \)\(27\!\cdots\!28\)\( T^{7} + \)\(22\!\cdots\!72\)\( T^{8} - \)\(27\!\cdots\!28\)\( p^{7} T^{9} - \)\(22\!\cdots\!10\)\( p^{14} T^{10} - \)\(34\!\cdots\!96\)\( p^{21} T^{11} + \)\(50\!\cdots\!09\)\( p^{28} T^{12} + \)\(31\!\cdots\!60\)\( p^{35} T^{13} + 27735348124934 p^{42} T^{14} - 16165212 p^{49} T^{15} + p^{56} T^{16} \) | |
| 97 | \( ( 1 + 1533112 T + 125450995201724 T^{2} + \)\(51\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!22\)\( T^{4} + \)\(51\!\cdots\!40\)\( p^{7} T^{5} + 125450995201724 p^{14} T^{6} + 1533112 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.85324645451158708520796775821, −5.79289972133568234878837077704, −5.68066459219110289519813673015, −5.17293440355572766243838462928, −4.84887499452842528784161981685, −4.75818220481150654180435142737, −4.65220978470592317773230291000, −4.62822736418493888411605649721, −4.34076495725300904493562310279, −3.63560191707945543740999589683, −3.58498752281438308537703711407, −3.50435125087861130262064421029, −3.42044708803573347549585798338, −3.40859789148693904180057548395, −2.76747436873652957787268933108, −2.45901420554922246862581032738, −2.31689925267250074097329590879, −2.25099980306108960463994722113, −1.71921532447988581587813440944, −1.70019530202614256167896501079, −1.18967230366982898565470394720, −1.03150034119275538849374066420, −0.875803950845976726133348610337, −0.795258230779835285083818683818, −0.43490637815226530920524083406, 0.43490637815226530920524083406, 0.795258230779835285083818683818, 0.875803950845976726133348610337, 1.03150034119275538849374066420, 1.18967230366982898565470394720, 1.70019530202614256167896501079, 1.71921532447988581587813440944, 2.25099980306108960463994722113, 2.31689925267250074097329590879, 2.45901420554922246862581032738, 2.76747436873652957787268933108, 3.40859789148693904180057548395, 3.42044708803573347549585798338, 3.50435125087861130262064421029, 3.58498752281438308537703711407, 3.63560191707945543740999589683, 4.34076495725300904493562310279, 4.62822736418493888411605649721, 4.65220978470592317773230291000, 4.75818220481150654180435142737, 4.84887499452842528784161981685, 5.17293440355572766243838462928, 5.68066459219110289519813673015, 5.79289972133568234878837077704, 5.85324645451158708520796775821
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.