Dirichlet series
| L(s) = 1 | − 12·3-s + 12·5-s − 14·7-s + 9·9-s − 88·11-s + 30·13-s − 144·15-s + 58·17-s − 190·19-s + 168·21-s − 184·23-s − 414·25-s + 432·27-s − 190·29-s − 60·31-s + 1.05e3·33-s − 168·35-s + 156·37-s − 360·39-s + 282·41-s − 810·43-s + 108·45-s + 564·47-s − 898·49-s − 696·51-s + 230·53-s − 1.05e3·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.07·5-s − 0.755·7-s + 1/3·9-s − 2.41·11-s + 0.640·13-s − 2.47·15-s + 0.827·17-s − 2.29·19-s + 1.74·21-s − 1.66·23-s − 3.31·25-s + 3.07·27-s − 1.21·29-s − 0.347·31-s + 5.57·33-s − 0.811·35-s + 0.693·37-s − 1.47·39-s + 1.07·41-s − 2.87·43-s + 0.357·45-s + 1.75·47-s − 2.61·49-s − 1.91·51-s + 0.596·53-s − 2.58·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{48} \cdot 23^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.23736\times 10^{15}\) |
| Root analytic conductor: | \(9.31937\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{48} \cdot 23^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 23 | \( ( 1 + p T )^{8} \) | |
| good | 3 | \( 1 + 4 p T + 5 p^{3} T^{2} + 40 p^{3} T^{3} + 965 p^{2} T^{4} + 6244 p^{2} T^{5} + 371930 T^{6} + 686036 p T^{7} + 3831022 p T^{8} + 686036 p^{4} T^{9} + 371930 p^{6} T^{10} + 6244 p^{11} T^{11} + 965 p^{14} T^{12} + 40 p^{18} T^{13} + 5 p^{21} T^{14} + 4 p^{22} T^{15} + p^{24} T^{16} \) |
| 5 | \( 1 - 12 T + 558 T^{2} - 6792 T^{3} + 171396 T^{4} - 1905872 T^{5} + 35327514 T^{6} - 342929052 T^{7} + 5210867942 T^{8} - 342929052 p^{3} T^{9} + 35327514 p^{6} T^{10} - 1905872 p^{9} T^{11} + 171396 p^{12} T^{12} - 6792 p^{15} T^{13} + 558 p^{18} T^{14} - 12 p^{21} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 + 2 p T + 1094 T^{2} + 3722 p T^{3} + 831492 T^{4} + 17743510 T^{5} + 458102450 T^{6} + 8894837758 T^{7} + 172843238790 T^{8} + 8894837758 p^{3} T^{9} + 458102450 p^{6} T^{10} + 17743510 p^{9} T^{11} + 831492 p^{12} T^{12} + 3722 p^{16} T^{13} + 1094 p^{18} T^{14} + 2 p^{22} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 + 8 p T + 8134 T^{2} + 527068 T^{3} + 29522140 T^{4} + 1476743188 T^{5} + 66227765058 T^{6} + 2673749809776 T^{7} + 103788190691478 T^{8} + 2673749809776 p^{3} T^{9} + 66227765058 p^{6} T^{10} + 1476743188 p^{9} T^{11} + 29522140 p^{12} T^{12} + 527068 p^{15} T^{13} + 8134 p^{18} T^{14} + 8 p^{22} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 30 T + 8623 T^{2} - 24962 p T^{3} + 38745457 T^{4} - 1778226588 T^{5} + 118720437366 T^{6} - 6007223069192 T^{7} + 287733902451010 T^{8} - 6007223069192 p^{3} T^{9} + 118720437366 p^{6} T^{10} - 1778226588 p^{9} T^{11} + 38745457 p^{12} T^{12} - 24962 p^{16} T^{13} + 8623 p^{18} T^{14} - 30 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 58 T + 24714 T^{2} - 739646 T^{3} + 265388300 T^{4} - 3013008610 T^{5} + 1862751072702 T^{6} - 3359705913894 T^{7} + 10169183353995862 T^{8} - 3359705913894 p^{3} T^{9} + 1862751072702 p^{6} T^{10} - 3013008610 p^{9} T^{11} + 265388300 p^{12} T^{12} - 739646 p^{15} T^{13} + 24714 p^{18} T^{14} - 58 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 10 p T + 46416 T^{2} + 5768430 T^{3} + 878213356 T^{4} + 86547684862 T^{5} + 10282296471280 T^{6} + 848607523793086 T^{7} + 83599454050384390 T^{8} + 848607523793086 p^{3} T^{9} + 10282296471280 p^{6} T^{10} + 86547684862 p^{9} T^{11} + 878213356 p^{12} T^{12} + 5768430 p^{15} T^{13} + 46416 p^{18} T^{14} + 10 p^{22} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 + 190 T + 81903 T^{2} + 12668238 T^{3} + 3348797321 T^{4} + 443232784680 T^{5} + 99790588535430 T^{6} + 11545386938443436 T^{7} + 86407098224178002 p T^{8} + 11545386938443436 p^{3} T^{9} + 99790588535430 p^{6} T^{10} + 443232784680 p^{9} T^{11} + 3348797321 p^{12} T^{12} + 12668238 p^{15} T^{13} + 81903 p^{18} T^{14} + 190 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 + 60 T + 88331 T^{2} - 212924 T^{3} + 3775918869 T^{4} - 218356055120 T^{5} + 126547839290226 T^{6} - 9799382872505480 T^{7} + 4033569334997275226 T^{8} - 9799382872505480 p^{3} T^{9} + 126547839290226 p^{6} T^{10} - 218356055120 p^{9} T^{11} + 3775918869 p^{12} T^{12} - 212924 p^{15} T^{13} + 88331 p^{18} T^{14} + 60 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 - 156 T + 305414 T^{2} - 48304008 T^{3} + 44269746164 T^{4} - 6510880736624 T^{5} + 3993208351902882 T^{6} - 511794502952013804 T^{7} + \)\(24\!\cdots\!30\)\( T^{8} - 511794502952013804 p^{3} T^{9} + 3993208351902882 p^{6} T^{10} - 6510880736624 p^{9} T^{11} + 44269746164 p^{12} T^{12} - 48304008 p^{15} T^{13} + 305414 p^{18} T^{14} - 156 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 282 T + 406967 T^{2} - 111531986 T^{3} + 79032706225 T^{4} - 19801930698600 T^{5} + 9602070331297422 T^{6} - 2091298025211225364 T^{7} + \)\(79\!\cdots\!82\)\( T^{8} - 2091298025211225364 p^{3} T^{9} + 9602070331297422 p^{6} T^{10} - 19801930698600 p^{9} T^{11} + 79032706225 p^{12} T^{12} - 111531986 p^{15} T^{13} + 406967 p^{18} T^{14} - 282 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 + 810 T + 748836 T^{2} + 400852146 T^{3} + 219302352916 T^{4} + 88392716782050 T^{5} + 35440160183076028 T^{6} + 11264147007704519274 T^{7} + \)\(35\!\cdots\!54\)\( T^{8} + 11264147007704519274 p^{3} T^{9} + 35440160183076028 p^{6} T^{10} + 88392716782050 p^{9} T^{11} + 219302352916 p^{12} T^{12} + 400852146 p^{15} T^{13} + 748836 p^{18} T^{14} + 810 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 - 12 p T + 357203 T^{2} - 119277468 T^{3} + 57768525893 T^{4} - 12628648199520 T^{5} + 4563365674672178 T^{6} - 578225878401858360 T^{7} + \)\(38\!\cdots\!70\)\( T^{8} - 578225878401858360 p^{3} T^{9} + 4563365674672178 p^{6} T^{10} - 12628648199520 p^{9} T^{11} + 57768525893 p^{12} T^{12} - 119277468 p^{15} T^{13} + 357203 p^{18} T^{14} - 12 p^{22} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 230 T + 570128 T^{2} - 141594314 T^{3} + 128952985292 T^{4} - 37826221869878 T^{5} + 14782294503604016 T^{6} - 6446592419167632138 T^{7} + \)\(14\!\cdots\!70\)\( T^{8} - 6446592419167632138 p^{3} T^{9} + 14782294503604016 p^{6} T^{10} - 37826221869878 p^{9} T^{11} + 128952985292 p^{12} T^{12} - 141594314 p^{15} T^{13} + 570128 p^{18} T^{14} - 230 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 + 1916 T + 2405664 T^{2} + 2321229932 T^{3} + 1839565138892 T^{4} + 1249730100885324 T^{5} + 747053166046109216 T^{6} + \)\(39\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!90\)\( T^{8} + \)\(39\!\cdots\!20\)\( p^{3} T^{9} + 747053166046109216 p^{6} T^{10} + 1249730100885324 p^{9} T^{11} + 1839565138892 p^{12} T^{12} + 2321229932 p^{15} T^{13} + 2405664 p^{18} T^{14} + 1916 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 22 T + 656584 T^{2} + 56665610 T^{3} + 289690058876 T^{4} + 25759150848710 T^{5} + 91312341450809528 T^{6} + 7654640322361851626 T^{7} + \)\(23\!\cdots\!26\)\( T^{8} + 7654640322361851626 p^{3} T^{9} + 91312341450809528 p^{6} T^{10} + 25759150848710 p^{9} T^{11} + 289690058876 p^{12} T^{12} + 56665610 p^{15} T^{13} + 656584 p^{18} T^{14} + 22 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 2292 T + 3793886 T^{2} + 4344144896 T^{3} + 4198586189676 T^{4} + 3337397232457784 T^{5} + 2382188962605544170 T^{6} + \)\(14\!\cdots\!60\)\( T^{7} + \)\(86\!\cdots\!26\)\( T^{8} + \)\(14\!\cdots\!60\)\( p^{3} T^{9} + 2382188962605544170 p^{6} T^{10} + 3337397232457784 p^{9} T^{11} + 4198586189676 p^{12} T^{12} + 4344144896 p^{15} T^{13} + 3793886 p^{18} T^{14} + 2292 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 2376 T + 3675615 T^{2} - 4279744976 T^{3} + 4188196909029 T^{4} - 3572345035677032 T^{5} + 2727549647711214498 T^{6} - \)\(18\!\cdots\!76\)\( T^{7} + \)\(11\!\cdots\!18\)\( T^{8} - \)\(18\!\cdots\!76\)\( p^{3} T^{9} + 2727549647711214498 p^{6} T^{10} - 3572345035677032 p^{9} T^{11} + 4188196909029 p^{12} T^{12} - 4279744976 p^{15} T^{13} + 3675615 p^{18} T^{14} - 2376 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 + 630 T + 2074167 T^{2} + 1112203230 T^{3} + 2126075486833 T^{4} + 984814771175032 T^{5} + 1400980936204462734 T^{6} + \)\(55\!\cdots\!32\)\( T^{7} + \)\(64\!\cdots\!82\)\( T^{8} + \)\(55\!\cdots\!32\)\( p^{3} T^{9} + 1400980936204462734 p^{6} T^{10} + 984814771175032 p^{9} T^{11} + 2126075486833 p^{12} T^{12} + 1112203230 p^{15} T^{13} + 2074167 p^{18} T^{14} + 630 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 892 T + 2843620 T^{2} - 1648787116 T^{3} + 3328960395652 T^{4} - 1293342518366300 T^{5} + 2388714781909334172 T^{6} - \)\(67\!\cdots\!32\)\( T^{7} + \)\(13\!\cdots\!78\)\( T^{8} - \)\(67\!\cdots\!32\)\( p^{3} T^{9} + 2388714781909334172 p^{6} T^{10} - 1293342518366300 p^{9} T^{11} + 3328960395652 p^{12} T^{12} - 1648787116 p^{15} T^{13} + 2843620 p^{18} T^{14} - 892 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 + 2800 T + 6734734 T^{2} + 11284901676 T^{3} + 16191548396524 T^{4} + 19398083944552692 T^{5} + 20218543045915497978 T^{6} + \)\(18\!\cdots\!52\)\( T^{7} + \)\(14\!\cdots\!06\)\( T^{8} + \)\(18\!\cdots\!52\)\( p^{3} T^{9} + 20218543045915497978 p^{6} T^{10} + 19398083944552692 p^{9} T^{11} + 16191548396524 p^{12} T^{12} + 11284901676 p^{15} T^{13} + 6734734 p^{18} T^{14} + 2800 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 + 308 T + 2580572 T^{2} + 537416492 T^{3} + 3737430388820 T^{4} + 466567404321124 T^{5} + 3957470808670947684 T^{6} + \)\(43\!\cdots\!64\)\( T^{7} + \)\(31\!\cdots\!90\)\( T^{8} + \)\(43\!\cdots\!64\)\( p^{3} T^{9} + 3957470808670947684 p^{6} T^{10} + 466567404321124 p^{9} T^{11} + 3737430388820 p^{12} T^{12} + 537416492 p^{15} T^{13} + 2580572 p^{18} T^{14} + 308 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 - 854 T + 3132426 T^{2} - 2694403618 T^{3} + 5557873802604 T^{4} - 4455992353790526 T^{5} + 7190741008072039166 T^{6} - \)\(51\!\cdots\!54\)\( T^{7} + \)\(72\!\cdots\!14\)\( T^{8} - \)\(51\!\cdots\!54\)\( p^{3} T^{9} + 7190741008072039166 p^{6} T^{10} - 4455992353790526 p^{9} T^{11} + 5557873802604 p^{12} T^{12} - 2694403618 p^{15} T^{13} + 3132426 p^{18} T^{14} - 854 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−4.03630209144009521237940898566, −3.92394145637932955475008160990, −3.76633788999028068553047269402, −3.68627005356804579590516419731, −3.67598911253509922370088794556, −3.61467878951172875445477524100, −3.37125560688354240068018535012, −3.35774937188101063063408846775, −3.15555996760764495784993465668, −2.72668353865807251308029044285, −2.69766523348999238617496922337, −2.60147672248414712687019702612, −2.54145687691683855120552076672, −2.50877504333493819021099835409, −2.36628810954434616002290239018, −2.33256478868911951244009084323, −1.97277089098738476305300643542, −1.80421188699432600531460994462, −1.67442640170556094134764104086, −1.49183817081434813619177605511, −1.41701300076627980877848852270, −1.26383031288071616136291189976, −1.25033350857530458615822580454, −0.954681742448365421501653938920, −0.62261733248286535705815709966, 0, 0, 0, 0, 0, 0, 0, 0, 0.62261733248286535705815709966, 0.954681742448365421501653938920, 1.25033350857530458615822580454, 1.26383031288071616136291189976, 1.41701300076627980877848852270, 1.49183817081434813619177605511, 1.67442640170556094134764104086, 1.80421188699432600531460994462, 1.97277089098738476305300643542, 2.33256478868911951244009084323, 2.36628810954434616002290239018, 2.50877504333493819021099835409, 2.54145687691683855120552076672, 2.60147672248414712687019702612, 2.69766523348999238617496922337, 2.72668353865807251308029044285, 3.15555996760764495784993465668, 3.35774937188101063063408846775, 3.37125560688354240068018535012, 3.61467878951172875445477524100, 3.67598911253509922370088794556, 3.68627005356804579590516419731, 3.76633788999028068553047269402, 3.92394145637932955475008160990, 4.03630209144009521237940898566
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.