Dirichlet series
| L(s) = 1 | − 4·3-s + 4·5-s − 4·7-s + 13·9-s − 6·11-s − 3·13-s − 16·15-s − 16·17-s − 4·19-s + 16·21-s − 5·23-s + 11·25-s − 29·27-s + 29-s + 11·31-s + 24·33-s − 16·35-s + 54·37-s + 12·39-s + 2·41-s − 11·43-s + 52·45-s + 7·47-s + 6·49-s + 64·51-s − 8·53-s − 24·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s − 1.51·7-s + 13/3·9-s − 1.80·11-s − 0.832·13-s − 4.13·15-s − 3.88·17-s − 0.917·19-s + 3.49·21-s − 1.04·23-s + 11/5·25-s − 5.58·27-s + 0.185·29-s + 1.97·31-s + 4.17·33-s − 2.70·35-s + 8.87·37-s + 1.92·39-s + 0.312·41-s − 1.67·43-s + 7.75·45-s + 1.02·47-s + 6/7·49-s + 8.96·51-s − 1.09·53-s − 3.23·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 3^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(68811.5\) |
| Root analytic conductor: | \(2.00610\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2922899116\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2922899116\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 4 T + p T^{2} - 11 T^{3} - 32 T^{4} - 11 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | |
| 7 | \( ( 1 + T + T^{2} )^{4} \) | |
| good | 5 | \( 1 - 4 T + p T^{2} + 18 T^{3} - 94 T^{4} + 232 T^{5} - 22 p T^{6} - 1011 T^{7} + 3826 T^{8} - 1011 p T^{9} - 22 p^{3} T^{10} + 232 p^{3} T^{11} - 94 p^{4} T^{12} + 18 p^{5} T^{13} + p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 6 T + T^{2} - 24 T^{3} + 49 T^{4} - 306 T^{5} - 2153 T^{6} - 1236 T^{7} + 7063 T^{8} - 1236 p T^{9} - 2153 p^{2} T^{10} - 306 p^{3} T^{11} + 49 p^{4} T^{12} - 24 p^{5} T^{13} + p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 3 T - 16 T^{2} + 3 p T^{3} + 337 T^{4} - 720 T^{5} + 2222 T^{6} + 906 p T^{7} - 3392 p T^{8} + 906 p^{2} T^{9} + 2222 p^{2} T^{10} - 720 p^{3} T^{11} + 337 p^{4} T^{12} + 3 p^{6} T^{13} - 16 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( ( 1 + 8 T + 35 T^{2} + 167 T^{3} + 886 T^{4} + 167 p T^{5} + 35 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 19 | \( ( 1 + 2 T + 55 T^{2} + 41 T^{3} + 1309 T^{4} + 41 p T^{5} + 55 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 + 5 T - 28 T^{2} - 177 T^{3} - 25 T^{4} - 494 T^{5} - 12137 T^{6} + 44331 T^{7} + 708646 T^{8} + 44331 p T^{9} - 12137 p^{2} T^{10} - 494 p^{3} T^{11} - 25 p^{4} T^{12} - 177 p^{5} T^{13} - 28 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - T - 49 T^{2} + 294 T^{3} + 1136 T^{4} - 11204 T^{5} + 49096 T^{6} + 227337 T^{7} - 2198075 T^{8} + 227337 p T^{9} + 49096 p^{2} T^{10} - 11204 p^{3} T^{11} + 1136 p^{4} T^{12} + 294 p^{5} T^{13} - 49 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 11 T - 39 T^{2} + 356 T^{3} + 5954 T^{4} - 25902 T^{5} - 215174 T^{6} + 3835 T^{7} + 11112081 T^{8} + 3835 p T^{9} - 215174 p^{2} T^{10} - 25902 p^{3} T^{11} + 5954 p^{4} T^{12} + 356 p^{5} T^{13} - 39 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 4104 p T^{5} + 412 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 - 2 T - 37 T^{2} + 432 T^{3} - 841 T^{4} - 13042 T^{5} + 100489 T^{6} + 72294 T^{7} - 3776099 T^{8} + 72294 p T^{9} + 100489 p^{2} T^{10} - 13042 p^{3} T^{11} - 841 p^{4} T^{12} + 432 p^{5} T^{13} - 37 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 11 T - 27 T^{2} - 596 T^{3} + 140 T^{4} + 2688 T^{5} - 152864 T^{6} + 330797 T^{7} + 13009029 T^{8} + 330797 p T^{9} - 152864 p^{2} T^{10} + 2688 p^{3} T^{11} + 140 p^{4} T^{12} - 596 p^{5} T^{13} - 27 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 7 T - 61 T^{2} - 174 T^{3} + 5108 T^{4} + 21898 T^{5} - 62918 T^{6} - 808359 T^{7} - 1538879 T^{8} - 808359 p T^{9} - 62918 p^{2} T^{10} + 21898 p^{3} T^{11} + 5108 p^{4} T^{12} - 174 p^{5} T^{13} - 61 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( ( 1 + 4 T + 83 T^{2} - 197 T^{3} + 2722 T^{4} - 197 p T^{5} + 83 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 - 9 T - 119 T^{2} + 1116 T^{3} + 10336 T^{4} - 79866 T^{5} - 564434 T^{6} + 2176353 T^{7} + 30131725 T^{8} + 2176353 p T^{9} - 564434 p^{2} T^{10} - 79866 p^{3} T^{11} + 10336 p^{4} T^{12} + 1116 p^{5} T^{13} - 119 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 7 T - 144 T^{2} - 1315 T^{3} + 11783 T^{4} + 112692 T^{5} - 470051 T^{6} - 3262925 T^{7} + 21742764 T^{8} - 3262925 p T^{9} - 470051 p^{2} T^{10} + 112692 p^{3} T^{11} + 11783 p^{4} T^{12} - 1315 p^{5} T^{13} - 144 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 12 T - 139 T^{2} - 1338 T^{3} + 22729 T^{4} + 133506 T^{5} - 1954153 T^{6} - 2368722 T^{7} + 170018875 T^{8} - 2368722 p T^{9} - 1954153 p^{2} T^{10} + 133506 p^{3} T^{11} + 22729 p^{4} T^{12} - 1338 p^{5} T^{13} - 139 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 + 12 T + 167 T^{2} + 591 T^{3} + 7587 T^{4} + 591 p T^{5} + 167 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( ( 1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 1504 p T^{5} + 232 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 79 | \( 1 + 22 T + 3 p T^{2} + 1616 T^{3} + 2882 T^{4} - 87342 T^{5} - 574370 T^{6} + 1050079 T^{7} + 29800944 T^{8} + 1050079 p T^{9} - 574370 p^{2} T^{10} - 87342 p^{3} T^{11} + 2882 p^{4} T^{12} + 1616 p^{5} T^{13} + 3 p^{7} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 6 T - 113 T^{2} - 1704 T^{3} + 3151 T^{4} + 166296 T^{5} + 1416277 T^{6} - 7918434 T^{7} - 172589093 T^{8} - 7918434 p T^{9} + 1416277 p^{2} T^{10} + 166296 p^{3} T^{11} + 3151 p^{4} T^{12} - 1704 p^{5} T^{13} - 113 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 14 T + 323 T^{2} + 2963 T^{3} + 41152 T^{4} + 2963 p T^{5} + 323 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + T - 123 T^{2} - 856 T^{3} - 4138 T^{4} + 70242 T^{5} + 167026 T^{6} - 386291 T^{7} + 107246157 T^{8} - 386291 p T^{9} + 167026 p^{2} T^{10} + 70242 p^{3} T^{11} - 4138 p^{4} T^{12} - 856 p^{5} T^{13} - 123 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.70919705391989326264456501014, −4.68089781117553579231906231438, −4.53958934574829593356853980847, −4.52493646615438735539138993794, −4.33657229461534922571340981070, −4.16846536921387515523386264965, −4.11242179736913214472946147129, −4.04372120918891461373073898039, −3.98856523118563174658757130706, −3.35368252505893746334792040417, −3.21925188628988643493125570530, −3.03041912759893241808004145792, −2.95453699176211040761489115804, −2.56974571564082249836451658942, −2.56514680581698784611717326997, −2.52191965799985794206669360522, −2.51426253395555334587232469480, −2.02801942974619115035422819389, −1.98587895397037711569186158025, −1.88042750161605822451069998510, −1.28060752794301789759996441900, −1.26701256605657189366694556429, −0.74306173748962589421243315522, −0.71719343507070073935222794465, −0.15038762162289920153033846570, 0.15038762162289920153033846570, 0.71719343507070073935222794465, 0.74306173748962589421243315522, 1.26701256605657189366694556429, 1.28060752794301789759996441900, 1.88042750161605822451069998510, 1.98587895397037711569186158025, 2.02801942974619115035422819389, 2.51426253395555334587232469480, 2.52191965799985794206669360522, 2.56514680581698784611717326997, 2.56974571564082249836451658942, 2.95453699176211040761489115804, 3.03041912759893241808004145792, 3.21925188628988643493125570530, 3.35368252505893746334792040417, 3.98856523118563174658757130706, 4.04372120918891461373073898039, 4.11242179736913214472946147129, 4.16846536921387515523386264965, 4.33657229461534922571340981070, 4.52493646615438735539138993794, 4.53958934574829593356853980847, 4.68089781117553579231906231438, 4.70919705391989326264456501014
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.