Dirichlet series
| L(s) = 1 | + 2-s + 2·4-s − 4·5-s − 2·7-s + 5·8-s − 4·10-s − 10·11-s + 5·13-s − 2·14-s + 8·16-s + 6·17-s + 8·19-s − 8·20-s − 10·22-s + 24·23-s − 4·25-s + 5·26-s − 4·28-s − 10·29-s + 18·31-s + 10·32-s + 6·34-s + 8·35-s + 8·38-s − 20·40-s − 5·41-s + 7·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 1.78·5-s − 0.755·7-s + 1.76·8-s − 1.26·10-s − 3.01·11-s + 1.38·13-s − 0.534·14-s + 2·16-s + 1.45·17-s + 1.83·19-s − 1.78·20-s − 2.13·22-s + 5.00·23-s − 4/5·25-s + 0.980·26-s − 0.755·28-s − 1.85·29-s + 3.23·31-s + 1.76·32-s + 1.02·34-s + 1.35·35-s + 1.29·38-s − 3.16·40-s − 0.780·41-s + 1.06·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{32} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(176555.\) |
| Root analytic conductor: | \(2.12779\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{32} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.245004659\) |
| \(L(\frac12)\) | \(\approx\) | \(2.245004659\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 2 T + T^{2} - 2 p T^{3} - 85 T^{4} - 2 p^{2} T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 - T - T^{2} - p T^{3} + T^{4} + 3 p T^{5} + 5 p T^{6} - p^{4} T^{7} - 7 T^{8} - p^{5} T^{9} + 5 p^{3} T^{10} + 3 p^{4} T^{11} + p^{4} T^{12} - p^{6} T^{13} - p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + 2 T + 8 T^{2} - 3 T^{3} + 9 T^{4} - 3 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( ( 1 + 5 T + 20 T^{2} - 9 T^{3} - 51 T^{4} - 9 p T^{5} + 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 - 5 T - 27 T^{2} + 90 T^{3} + 822 T^{4} - 1305 T^{5} - 15538 T^{6} + 5450 T^{7} + 240237 T^{8} + 5450 p T^{9} - 15538 p^{2} T^{10} - 1305 p^{3} T^{11} + 822 p^{4} T^{12} + 90 p^{5} T^{13} - 27 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 6 T + 13 T^{2} - 234 T^{3} + 1180 T^{4} - 2436 T^{5} + 28843 T^{6} - 124320 T^{7} + 209935 T^{8} - 124320 p T^{9} + 28843 p^{2} T^{10} - 2436 p^{3} T^{11} + 1180 p^{4} T^{12} - 234 p^{5} T^{13} + 13 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 8 T - 12 T^{2} + 210 T^{3} + 387 T^{4} - 4983 T^{5} - 907 T^{6} + 40979 T^{7} - 34155 T^{8} + 40979 p T^{9} - 907 p^{2} T^{10} - 4983 p^{3} T^{11} + 387 p^{4} T^{12} + 210 p^{5} T^{13} - 12 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 12 T + 128 T^{2} - 861 T^{3} + 4839 T^{4} - 861 p T^{5} + 128 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 10 T - 16 T^{2} - 400 T^{3} + 547 T^{4} + 10920 T^{5} - 15332 T^{6} - 43370 T^{7} + 1312304 T^{8} - 43370 p T^{9} - 15332 p^{2} T^{10} + 10920 p^{3} T^{11} + 547 p^{4} T^{12} - 400 p^{5} T^{13} - 16 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 18 T + 107 T^{2} - 286 T^{3} + 2998 T^{4} - 34484 T^{5} + 185307 T^{6} - 773074 T^{7} + 4072539 T^{8} - 773074 p T^{9} + 185307 p^{2} T^{10} - 34484 p^{3} T^{11} + 2998 p^{4} T^{12} - 286 p^{5} T^{13} + 107 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 70 T^{2} + 2 p T^{3} + 1507 T^{4} - 107 p T^{5} - 44481 T^{6} + 1787 p T^{7} + 2879535 T^{8} + 1787 p^{2} T^{9} - 44481 p^{2} T^{10} - 107 p^{4} T^{11} + 1507 p^{4} T^{12} + 2 p^{6} T^{13} - 70 p^{6} T^{14} + p^{8} T^{16} \) | |
| 41 | \( 1 + 5 T - 67 T^{2} - 518 T^{3} + 1396 T^{4} + 18807 T^{5} + 22864 T^{6} - 310504 T^{7} - 2039527 T^{8} - 310504 p T^{9} + 22864 p^{2} T^{10} + 18807 p^{3} T^{11} + 1396 p^{4} T^{12} - 518 p^{5} T^{13} - 67 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 7 T - 93 T^{2} + 460 T^{3} + 6512 T^{4} - 14613 T^{5} - 381128 T^{6} + 258368 T^{7} + 17687871 T^{8} + 258368 p T^{9} - 381128 p^{2} T^{10} - 14613 p^{3} T^{11} + 6512 p^{4} T^{12} + 460 p^{5} T^{13} - 93 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 21 T + 100 T^{2} + 345 T^{3} + 16636 T^{4} + 143070 T^{5} + 183049 T^{6} + 4445316 T^{7} + 72265597 T^{8} + 4445316 p T^{9} + 183049 p^{2} T^{10} + 143070 p^{3} T^{11} + 16636 p^{4} T^{12} + 345 p^{5} T^{13} + 100 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 12 T - 68 T^{2} - 1062 T^{3} + 6535 T^{4} + 75243 T^{5} - 260759 T^{6} - 1342677 T^{7} + 17779051 T^{8} - 1342677 p T^{9} - 260759 p^{2} T^{10} + 75243 p^{3} T^{11} + 6535 p^{4} T^{12} - 1062 p^{5} T^{13} - 68 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 6 T - 92 T^{2} + 954 T^{3} + 2887 T^{4} - 61035 T^{5} + 261295 T^{6} + 1925871 T^{7} - 28695155 T^{8} + 1925871 p T^{9} + 261295 p^{2} T^{10} - 61035 p^{3} T^{11} + 2887 p^{4} T^{12} + 954 p^{5} T^{13} - 92 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 20 T + 129 T^{2} + 600 T^{3} - 13422 T^{4} + 67200 T^{5} + 179009 T^{6} - 5446450 T^{7} + 52714467 T^{8} - 5446450 p T^{9} + 179009 p^{2} T^{10} + 67200 p^{3} T^{11} - 13422 p^{4} T^{12} + 600 p^{5} T^{13} + 129 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 5 T - 171 T^{2} + 882 T^{3} + 16122 T^{4} - 68463 T^{5} - 1142680 T^{6} + 2126840 T^{7} + 75498939 T^{8} + 2126840 p T^{9} - 1142680 p^{2} T^{10} - 68463 p^{3} T^{11} + 16122 p^{4} T^{12} + 882 p^{5} T^{13} - 171 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 - 9 T + 257 T^{2} - 1782 T^{3} + 26655 T^{4} - 1782 p T^{5} + 257 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 - 6 T - 106 T^{2} + 1922 T^{3} - 521 T^{4} - 149849 T^{5} + 1074153 T^{6} + 64841 p T^{7} - 95434341 T^{8} + 64841 p^{2} T^{9} + 1074153 p^{2} T^{10} - 149849 p^{3} T^{11} - 521 p^{4} T^{12} + 1922 p^{5} T^{13} - 106 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 10 T - 132 T^{2} + 2410 T^{3} + 5993 T^{4} - 251895 T^{5} + 1010707 T^{6} + 10093865 T^{7} - 133401879 T^{8} + 10093865 p T^{9} + 1010707 p^{2} T^{10} - 251895 p^{3} T^{11} + 5993 p^{4} T^{12} + 2410 p^{5} T^{13} - 132 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 9 T - 125 T^{2} - 792 T^{3} + 22342 T^{4} + 122211 T^{5} - 431318 T^{6} - 7892478 T^{7} - 36023969 T^{8} - 7892478 p T^{9} - 431318 p^{2} T^{10} + 122211 p^{3} T^{11} + 22342 p^{4} T^{12} - 792 p^{5} T^{13} - 125 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 22 T + 14 T^{2} - 1510 T^{3} + 22753 T^{4} + 329007 T^{5} - 914375 T^{6} + 3424183 T^{7} + 341951435 T^{8} + 3424183 p T^{9} - 914375 p^{2} T^{10} + 329007 p^{3} T^{11} + 22753 p^{4} T^{12} - 1510 p^{5} T^{13} + 14 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 9 T - 49 T^{2} - 844 T^{3} + 6154 T^{4} + 130021 T^{5} + 1124340 T^{6} - 16624204 T^{7} - 16165707 T^{8} - 16624204 p T^{9} + 1124340 p^{2} T^{10} + 130021 p^{3} T^{11} + 6154 p^{4} T^{12} - 844 p^{5} T^{13} - 49 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.81249659076999563898275107065, −4.65533219899774101459832209530, −4.47576240646190584106715399072, −4.28733347312209697012685290279, −4.15655023255511653442291871524, −3.76576738162387310383374905924, −3.67712183166590233647179654609, −3.64715519694493897332570798031, −3.63071076076682206717339498288, −3.52889758715954848751719188489, −3.11202772858177296534480529033, −3.09188054599004707205348210031, −3.07963731883049288914807396667, −3.05012483543573953788952574420, −2.73099348122642067287522960671, −2.32335652869441831762866935630, −2.28498999817861267102498933794, −2.27162657645311560482111892776, −2.14965643347262131619169226967, −1.46736208676832513440033444857, −1.24371700920535322174189681782, −1.16227743366991353231271420717, −1.11242981484150544116224680527, −0.821993535332624826612743601917, −0.20488136759966419623747573346, 0.20488136759966419623747573346, 0.821993535332624826612743601917, 1.11242981484150544116224680527, 1.16227743366991353231271420717, 1.24371700920535322174189681782, 1.46736208676832513440033444857, 2.14965643347262131619169226967, 2.27162657645311560482111892776, 2.28498999817861267102498933794, 2.32335652869441831762866935630, 2.73099348122642067287522960671, 3.05012483543573953788952574420, 3.07963731883049288914807396667, 3.09188054599004707205348210031, 3.11202772858177296534480529033, 3.52889758715954848751719188489, 3.63071076076682206717339498288, 3.64715519694493897332570798031, 3.67712183166590233647179654609, 3.76576738162387310383374905924, 4.15655023255511653442291871524, 4.28733347312209697012685290279, 4.47576240646190584106715399072, 4.65533219899774101459832209530, 4.81249659076999563898275107065
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.