Dirichlet series
| L(s) = 1 | − 3·2-s + 2·4-s − 8·5-s + 9·7-s + 2·8-s + 24·10-s − 8·11-s + 10·13-s − 27·14-s − 3·16-s − 8·17-s + 7·19-s − 16·20-s + 24·22-s − 6·23-s + 36·25-s − 30·26-s + 18·28-s − 6·29-s + 9·31-s + 24·34-s − 72·35-s + 3·37-s − 21·38-s − 16·40-s + 16·41-s + 30·43-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 4-s − 3.57·5-s + 3.40·7-s + 0.707·8-s + 7.58·10-s − 2.41·11-s + 2.77·13-s − 7.21·14-s − 3/4·16-s − 1.94·17-s + 1.60·19-s − 3.57·20-s + 5.11·22-s − 1.25·23-s + 36/5·25-s − 5.88·26-s + 3.40·28-s − 1.11·29-s + 1.61·31-s + 4.11·34-s − 12.1·35-s + 0.493·37-s − 3.40·38-s − 2.52·40-s + 2.49·41-s + 4.57·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 5^{8} \cdot 11^{8} \cdot 17^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.15575\times 10^{14}\) |
| Root analytic conductor: | \(8.19720\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 5^{8} \cdot 11^{8} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.101576327\) |
| \(L(\frac12)\) | \(\approx\) | \(5.101576327\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{8} \) | |
| 11 | \( ( 1 + T )^{8} \) | |
| 17 | \( ( 1 + T )^{8} \) | |
| good | 2 | \( 1 + 3 T + 7 T^{2} + 13 T^{3} + 11 p T^{4} + 35 T^{5} + 57 T^{6} + 91 T^{7} + 129 T^{8} + 91 p T^{9} + 57 p^{2} T^{10} + 35 p^{3} T^{11} + 11 p^{5} T^{12} + 13 p^{5} T^{13} + 7 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 9 T + 9 p T^{2} - 328 T^{3} + 1479 T^{4} - 5682 T^{5} + 19686 T^{6} - 60427 T^{7} + 168822 T^{8} - 60427 p T^{9} + 19686 p^{2} T^{10} - 5682 p^{3} T^{11} + 1479 p^{4} T^{12} - 328 p^{5} T^{13} + 9 p^{7} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 10 T + 103 T^{2} - 649 T^{3} + 4111 T^{4} - 19878 T^{5} + 96410 T^{6} - 380651 T^{7} + 1508446 T^{8} - 380651 p T^{9} + 96410 p^{2} T^{10} - 19878 p^{3} T^{11} + 4111 p^{4} T^{12} - 649 p^{5} T^{13} + 103 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 7 T + 89 T^{2} - 422 T^{3} + 3715 T^{4} - 14832 T^{5} + 106686 T^{6} - 359773 T^{7} + 2261330 T^{8} - 359773 p T^{9} + 106686 p^{2} T^{10} - 14832 p^{3} T^{11} + 3715 p^{4} T^{12} - 422 p^{5} T^{13} + 89 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 6 T + 102 T^{2} + 508 T^{3} + 250 p T^{4} + 25018 T^{5} + 213491 T^{6} + 808828 T^{7} + 5748560 T^{8} + 808828 p T^{9} + 213491 p^{2} T^{10} + 25018 p^{3} T^{11} + 250 p^{5} T^{12} + 508 p^{5} T^{13} + 102 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 6 T + 112 T^{2} + 444 T^{3} + 5323 T^{4} + 13416 T^{5} + 174868 T^{6} + 307646 T^{7} + 5239856 T^{8} + 307646 p T^{9} + 174868 p^{2} T^{10} + 13416 p^{3} T^{11} + 5323 p^{4} T^{12} + 444 p^{5} T^{13} + 112 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 9 T + 255 T^{2} - 1840 T^{3} + 28023 T^{4} - 164490 T^{5} + 1746582 T^{6} - 8333059 T^{7} + 67773990 T^{8} - 8333059 p T^{9} + 1746582 p^{2} T^{10} - 164490 p^{3} T^{11} + 28023 p^{4} T^{12} - 1840 p^{5} T^{13} + 255 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 3 T + 173 T^{2} - 456 T^{3} + 14479 T^{4} - 38200 T^{5} + 816190 T^{6} - 2150397 T^{7} + 34516250 T^{8} - 2150397 p T^{9} + 816190 p^{2} T^{10} - 38200 p^{3} T^{11} + 14479 p^{4} T^{12} - 456 p^{5} T^{13} + 173 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 16 T + 270 T^{2} - 2707 T^{3} + 28897 T^{4} - 226559 T^{5} + 1897026 T^{6} - 12490150 T^{7} + 89182300 T^{8} - 12490150 p T^{9} + 1897026 p^{2} T^{10} - 226559 p^{3} T^{11} + 28897 p^{4} T^{12} - 2707 p^{5} T^{13} + 270 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 30 T + 585 T^{2} - 190 p T^{3} + 94469 T^{4} - 924874 T^{5} + 8036395 T^{6} - 61856158 T^{7} + 429378092 T^{8} - 61856158 p T^{9} + 8036395 p^{2} T^{10} - 924874 p^{3} T^{11} + 94469 p^{4} T^{12} - 190 p^{6} T^{13} + 585 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 9 T + 199 T^{2} + 1284 T^{3} + 20625 T^{4} + 117358 T^{5} + 1502061 T^{6} + 7261491 T^{7} + 79752564 T^{8} + 7261491 p T^{9} + 1502061 p^{2} T^{10} + 117358 p^{3} T^{11} + 20625 p^{4} T^{12} + 1284 p^{5} T^{13} + 199 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 2 T + 84 T^{2} - 119 T^{3} + 5697 T^{4} + 4477 T^{5} + 361788 T^{6} - 63192 T^{7} + 21860524 T^{8} - 63192 p T^{9} + 361788 p^{2} T^{10} + 4477 p^{3} T^{11} + 5697 p^{4} T^{12} - 119 p^{5} T^{13} + 84 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 6 T + 176 T^{2} - 1247 T^{3} + 20959 T^{4} - 145293 T^{5} + 1778720 T^{6} - 11950392 T^{7} + 117598256 T^{8} - 11950392 p T^{9} + 1778720 p^{2} T^{10} - 145293 p^{3} T^{11} + 20959 p^{4} T^{12} - 1247 p^{5} T^{13} + 176 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - T + 234 T^{2} - 469 T^{3} + 30068 T^{4} - 95450 T^{5} + 2630435 T^{6} - 9851120 T^{7} + 178365868 T^{8} - 9851120 p T^{9} + 2630435 p^{2} T^{10} - 95450 p^{3} T^{11} + 30068 p^{4} T^{12} - 469 p^{5} T^{13} + 234 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 34 T + 868 T^{2} - 14402 T^{3} + 200410 T^{4} - 2161090 T^{5} + 21116779 T^{6} - 177430698 T^{7} + 1507352660 T^{8} - 177430698 p T^{9} + 21116779 p^{2} T^{10} - 2161090 p^{3} T^{11} + 200410 p^{4} T^{12} - 14402 p^{5} T^{13} + 868 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 19 T + 430 T^{2} + 6157 T^{3} + 88399 T^{4} + 993633 T^{5} + 11156178 T^{6} + 103580407 T^{7} + 949837504 T^{8} + 103580407 p T^{9} + 11156178 p^{2} T^{10} + 993633 p^{3} T^{11} + 88399 p^{4} T^{12} + 6157 p^{5} T^{13} + 430 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 20 T + 390 T^{2} - 5512 T^{3} + 71769 T^{4} - 796592 T^{5} + 8644710 T^{6} - 81826068 T^{7} + 743830420 T^{8} - 81826068 p T^{9} + 8644710 p^{2} T^{10} - 796592 p^{3} T^{11} + 71769 p^{4} T^{12} - 5512 p^{5} T^{13} + 390 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 6 T + 116 T^{2} + 216 T^{3} + 18839 T^{4} + 38812 T^{5} + 1410400 T^{6} + 564330 T^{7} + 145898472 T^{8} + 564330 p T^{9} + 1410400 p^{2} T^{10} + 38812 p^{3} T^{11} + 18839 p^{4} T^{12} + 216 p^{5} T^{13} + 116 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 23 T + 625 T^{2} + 8840 T^{3} + 136903 T^{4} + 1374058 T^{5} + 15964194 T^{6} + 130158125 T^{7} + 1383899762 T^{8} + 130158125 p T^{9} + 15964194 p^{2} T^{10} + 1374058 p^{3} T^{11} + 136903 p^{4} T^{12} + 8840 p^{5} T^{13} + 625 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 16 T + 381 T^{2} - 3172 T^{3} + 56235 T^{4} - 382508 T^{5} + 7312399 T^{6} - 49537344 T^{7} + 790482704 T^{8} - 49537344 p T^{9} + 7312399 p^{2} T^{10} - 382508 p^{3} T^{11} + 56235 p^{4} T^{12} - 3172 p^{5} T^{13} + 381 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 9 T + 611 T^{2} - 4542 T^{3} + 174703 T^{4} - 1089108 T^{5} + 30516246 T^{6} - 159738169 T^{7} + 3571979222 T^{8} - 159738169 p T^{9} + 30516246 p^{2} T^{10} - 1089108 p^{3} T^{11} + 174703 p^{4} T^{12} - 4542 p^{5} T^{13} + 611 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.13272005496147476513237173013, −3.10977582201903643582665637915, −2.84073976120353485273306939730, −2.80482430627000854239280927413, −2.65297024009190870406387109627, −2.64862576625801925131840015928, −2.39134661464668008150451501835, −2.29751195374221605958620363913, −2.28415362643885645661459449734, −2.23001630110059112007453245431, −2.09747373198813260351055248637, −1.72667623020476062210374923972, −1.63938382297761949530778794651, −1.53220915882067479231650473819, −1.53138124486569097205731777407, −1.43113659156506463105802172664, −1.40079237934833319211867153005, −1.03629074506500647573295751089, −0.75413792458992987810155993365, −0.75018502160536729984420943489, −0.64542109227340896865099409561, −0.52651864406017512383520039537, −0.48171336892617145189122445486, −0.38537590872564937810284530903, −0.37699459152592337715409284844, 0.37699459152592337715409284844, 0.38537590872564937810284530903, 0.48171336892617145189122445486, 0.52651864406017512383520039537, 0.64542109227340896865099409561, 0.75018502160536729984420943489, 0.75413792458992987810155993365, 1.03629074506500647573295751089, 1.40079237934833319211867153005, 1.43113659156506463105802172664, 1.53138124486569097205731777407, 1.53220915882067479231650473819, 1.63938382297761949530778794651, 1.72667623020476062210374923972, 2.09747373198813260351055248637, 2.23001630110059112007453245431, 2.28415362643885645661459449734, 2.29751195374221605958620363913, 2.39134661464668008150451501835, 2.64862576625801925131840015928, 2.65297024009190870406387109627, 2.80482430627000854239280927413, 2.84073976120353485273306939730, 3.10977582201903643582665637915, 3.13272005496147476513237173013
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.