Dirichlet series
| L(s) = 1 | − 16·5-s − 32·7-s + 52·9-s − 80·11-s − 28·13-s − 8·17-s − 56·19-s − 624·23-s + 354·25-s − 144·27-s − 216·29-s + 528·31-s + 512·35-s + 96·37-s + 552·41-s − 112·43-s − 832·45-s − 304·47-s + 1.19e3·49-s + 1.31e3·53-s + 1.28e3·55-s + 224·59-s + 2.66e3·61-s − 1.66e3·63-s + 448·65-s − 272·67-s + 1.12e3·71-s + ⋯ |
| L(s) = 1 | − 1.43·5-s − 1.72·7-s + 1.92·9-s − 2.19·11-s − 0.597·13-s − 0.114·17-s − 0.676·19-s − 5.65·23-s + 2.83·25-s − 1.02·27-s − 1.38·29-s + 3.05·31-s + 2.47·35-s + 0.426·37-s + 2.10·41-s − 0.397·43-s − 2.75·45-s − 0.943·47-s + 3.48·49-s + 3.41·53-s + 3.13·55-s + 0.494·59-s + 5.59·61-s − 3.32·63-s + 0.854·65-s − 0.495·67-s + 1.87·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.20922\times 10^{6}\) |
| Root analytic conductor: | \(2.70485\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 31^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(3.141300642\) |
| \(L(\frac12)\) | \(\approx\) | \(3.141300642\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 - 528 T + 171920 T^{2} - 1347312 p T^{3} + 8229342 p^{2} T^{4} - 1347312 p^{4} T^{5} + 171920 p^{6} T^{6} - 528 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 3 | \( 1 - 52 T^{2} + 16 p^{2} T^{3} + 433 p T^{4} - 632 p^{2} T^{5} + 7940 T^{6} + 11224 p^{2} T^{7} - 668600 T^{8} + 11224 p^{5} T^{9} + 7940 p^{6} T^{10} - 632 p^{11} T^{11} + 433 p^{13} T^{12} + 16 p^{17} T^{13} - 52 p^{18} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 + 16 T - 98 T^{2} + 448 T^{3} + 40033 T^{4} - 144528 T^{5} - 2770306 T^{6} + 506688 p T^{7} - 6214924 p^{2} T^{8} + 506688 p^{4} T^{9} - 2770306 p^{6} T^{10} - 144528 p^{9} T^{11} + 40033 p^{12} T^{12} + 448 p^{15} T^{13} - 98 p^{18} T^{14} + 16 p^{21} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 + 32 T - 172 T^{2} - 23872 T^{3} - 158077 T^{4} + 1398816 p T^{5} + 195189276 T^{6} - 1236590272 T^{7} - 77579474440 T^{8} - 1236590272 p^{3} T^{9} + 195189276 p^{6} T^{10} + 1398816 p^{10} T^{11} - 158077 p^{12} T^{12} - 23872 p^{15} T^{13} - 172 p^{18} T^{14} + 32 p^{21} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 + 80 T - 36 p T^{2} - 8816 p T^{3} + 6080747 T^{4} + 286725016 T^{5} - 5816203428 T^{6} - 484554744 p T^{7} + 151069562040 p^{2} T^{8} - 484554744 p^{4} T^{9} - 5816203428 p^{6} T^{10} + 286725016 p^{9} T^{11} + 6080747 p^{12} T^{12} - 8816 p^{16} T^{13} - 36 p^{19} T^{14} + 80 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 + 28 T - 1346 T^{2} + 146200 T^{3} + 7581609 T^{4} - 84631896 T^{5} + 29001788190 T^{6} + 1069190483124 T^{7} - 29324900091356 T^{8} + 1069190483124 p^{3} T^{9} + 29001788190 p^{6} T^{10} - 84631896 p^{9} T^{11} + 7581609 p^{12} T^{12} + 146200 p^{15} T^{13} - 1346 p^{18} T^{14} + 28 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 + 8 T - 1066 p T^{2} - 108432 T^{3} + 199184249 T^{4} + 844158368 T^{5} - 1484770244634 T^{6} - 1654594813592 T^{7} + 8408587588284820 T^{8} - 1654594813592 p^{3} T^{9} - 1484770244634 p^{6} T^{10} + 844158368 p^{9} T^{11} + 199184249 p^{12} T^{12} - 108432 p^{15} T^{13} - 1066 p^{19} T^{14} + 8 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 56 T - 17476 T^{2} - 77968 T^{3} + 210507875 T^{4} - 3217089768 T^{5} - 1624574380524 T^{6} + 472023220064 p T^{7} + 26490496278824 p^{2} T^{8} + 472023220064 p^{4} T^{9} - 1624574380524 p^{6} T^{10} - 3217089768 p^{9} T^{11} + 210507875 p^{12} T^{12} - 77968 p^{15} T^{13} - 17476 p^{18} T^{14} + 56 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( ( 1 + 312 T + 53036 T^{2} + 279592 p T^{3} + 715289094 T^{4} + 279592 p^{4} T^{5} + 53036 p^{6} T^{6} + 312 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 29 | \( ( 1 + 108 T + 43944 T^{2} + 5913972 T^{3} + 1592569246 T^{4} + 5913972 p^{3} T^{5} + 43944 p^{6} T^{6} + 108 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 37 | \( 1 - 96 T - 145234 T^{2} + 7034016 T^{3} + 12356587505 T^{4} - 235716180240 T^{5} - 754950052404690 T^{6} + 170179243236816 p T^{7} + 27563696107546580 p^{2} T^{8} + 170179243236816 p^{4} T^{9} - 754950052404690 p^{6} T^{10} - 235716180240 p^{9} T^{11} + 12356587505 p^{12} T^{12} + 7034016 p^{15} T^{13} - 145234 p^{18} T^{14} - 96 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 552 T + 111846 T^{2} + 11443984 T^{3} - 11974819351 T^{4} + 2925435776960 T^{5} - 142237995047114 T^{6} - 99500379023649128 T^{7} + 37251840382810522708 T^{8} - 99500379023649128 p^{3} T^{9} - 142237995047114 p^{6} T^{10} + 2925435776960 p^{9} T^{11} - 11974819351 p^{12} T^{12} + 11443984 p^{15} T^{13} + 111846 p^{18} T^{14} - 552 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 + 112 T - 274260 T^{2} - 23413072 T^{3} + 46182376131 T^{4} + 2787583760600 T^{5} - 5304504317450908 T^{6} - 91682544147093192 T^{7} + \)\(47\!\cdots\!24\)\( T^{8} - 91682544147093192 p^{3} T^{9} - 5304504317450908 p^{6} T^{10} + 2787583760600 p^{9} T^{11} + 46182376131 p^{12} T^{12} - 23413072 p^{15} T^{13} - 274260 p^{18} T^{14} + 112 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( ( 1 + 152 T + 83596 T^{2} + 15254968 T^{3} + 9723735206 T^{4} + 15254968 p^{3} T^{5} + 83596 p^{6} T^{6} + 152 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 53 | \( 1 - 1316 T + 618846 T^{2} - 211989288 T^{3} + 136054448553 T^{4} - 57550952803960 T^{5} + 9047989090716414 T^{6} - 3632289922682087884 T^{7} + \)\(26\!\cdots\!04\)\( T^{8} - 3632289922682087884 p^{3} T^{9} + 9047989090716414 p^{6} T^{10} - 57550952803960 p^{9} T^{11} + 136054448553 p^{12} T^{12} - 211989288 p^{15} T^{13} + 618846 p^{18} T^{14} - 1316 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 224 T - 251900 T^{2} + 91183984 T^{3} - 36539733029 T^{4} + 8974990473912 T^{5} - 5090133621063220 T^{6} - 3676235401400234424 T^{7} + \)\(60\!\cdots\!48\)\( T^{8} - 3676235401400234424 p^{3} T^{9} - 5090133621063220 p^{6} T^{10} + 8974990473912 p^{9} T^{11} - 36539733029 p^{12} T^{12} + 91183984 p^{15} T^{13} - 251900 p^{18} T^{14} - 224 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( ( 1 - 1332 T + 1395736 T^{2} - 959173740 T^{3} + 531295388606 T^{4} - 959173740 p^{3} T^{5} + 1395736 p^{6} T^{6} - 1332 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 67 | \( 1 + 272 T - 952764 T^{2} - 136392752 T^{3} + 562711156715 T^{4} + 39278693394600 T^{5} - 234655439823068852 T^{6} - 5297124990757102840 T^{7} + \)\(76\!\cdots\!52\)\( T^{8} - 5297124990757102840 p^{3} T^{9} - 234655439823068852 p^{6} T^{10} + 39278693394600 p^{9} T^{11} + 562711156715 p^{12} T^{12} - 136392752 p^{15} T^{13} - 952764 p^{18} T^{14} + 272 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 1120 T + 25452 T^{2} + 5386816 p T^{3} - 124556074693 T^{4} + 60209130132448 T^{5} - 62941879516097916 T^{6} - 25561406415022248768 T^{7} + \)\(50\!\cdots\!00\)\( T^{8} - 25561406415022248768 p^{3} T^{9} - 62941879516097916 p^{6} T^{10} + 60209130132448 p^{9} T^{11} - 124556074693 p^{12} T^{12} + 5386816 p^{16} T^{13} + 25452 p^{18} T^{14} - 1120 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 248 T - 637994 T^{2} + 303519152 T^{3} + 128942693913 T^{4} - 110209038297408 T^{5} + 34893644575309574 T^{6} + 17108070728583606600 T^{7} - \)\(21\!\cdots\!92\)\( T^{8} + 17108070728583606600 p^{3} T^{9} + 34893644575309574 p^{6} T^{10} - 110209038297408 p^{9} T^{11} + 128942693913 p^{12} T^{12} + 303519152 p^{15} T^{13} - 637994 p^{18} T^{14} - 248 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 824 T - 361148 T^{2} - 210510016 T^{3} + 455772967395 T^{4} + 97159703203488 T^{5} + 104715839079429164 T^{6} - 91816979794714083672 T^{7} - \)\(81\!\cdots\!28\)\( T^{8} - 91816979794714083672 p^{3} T^{9} + 104715839079429164 p^{6} T^{10} + 97159703203488 p^{9} T^{11} + 455772967395 p^{12} T^{12} - 210510016 p^{15} T^{13} - 361148 p^{18} T^{14} - 824 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 - 1616 T + 14252 T^{2} + 994804176 T^{3} + 241096290963 T^{4} - 762769818434264 T^{5} + 40712455231954980 T^{6} + 41020956880394678696 T^{7} + \)\(14\!\cdots\!60\)\( T^{8} + 41020956880394678696 p^{3} T^{9} + 40712455231954980 p^{6} T^{10} - 762769818434264 p^{9} T^{11} + 241096290963 p^{12} T^{12} + 994804176 p^{15} T^{13} + 14252 p^{18} T^{14} - 1616 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( ( 1 - 92 T + 2170904 T^{2} - 178243412 T^{3} + 2142527544654 T^{4} - 178243412 p^{3} T^{5} + 2170904 p^{6} T^{6} - 92 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 97 | \( ( 1 - 1260 T + 2135944 T^{2} - 1895841636 T^{3} + 1913432896526 T^{4} - 1895841636 p^{3} T^{5} + 2135944 p^{6} T^{6} - 1260 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
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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
−5.69403026118911392852006859693, −5.51843721400602589913421606960, −5.30109147693904902881326298805, −5.28295103947672719513321227900, −5.01623428591318209391099177683, −4.75045197872121455034122029029, −4.54314585151802371822643359385, −4.23656054743975398889359044326, −4.15704852905073919447282447858, −3.95157916287794395290831023090, −3.91822310402080053081146046794, −3.86122002095395105524031930338, −3.79247556163602705744408780662, −3.33641940741117546267600430054, −2.99002063734964246395200453819, −2.66828833044680183568084751702, −2.61970938618570049748937967882, −2.45479044751172857923500635965, −1.99309348790632224591525628866, −1.96700783369464821996074467744, −1.95858929785534846675634059586, −0.901475236808122727282522924743, −0.57364634583737444829608012968, −0.55450633600783568178269452658, −0.46766880413749423594827292789, 0.46766880413749423594827292789, 0.55450633600783568178269452658, 0.57364634583737444829608012968, 0.901475236808122727282522924743, 1.95858929785534846675634059586, 1.96700783369464821996074467744, 1.99309348790632224591525628866, 2.45479044751172857923500635965, 2.61970938618570049748937967882, 2.66828833044680183568084751702, 2.99002063734964246395200453819, 3.33641940741117546267600430054, 3.79247556163602705744408780662, 3.86122002095395105524031930338, 3.91822310402080053081146046794, 3.95157916287794395290831023090, 4.15704852905073919447282447858, 4.23656054743975398889359044326, 4.54314585151802371822643359385, 4.75045197872121455034122029029, 5.01623428591318209391099177683, 5.28295103947672719513321227900, 5.30109147693904902881326298805, 5.51843721400602589913421606960, 5.69403026118911392852006859693
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.