Dirichlet series
| L(s) = 1 | + 2-s − 3·4-s − 12·7-s − 3·8-s − 12·11-s − 14·13-s − 12·14-s + 4·16-s − 17-s + 16·19-s − 12·22-s − 4·23-s − 14·26-s + 36·28-s − 2·29-s + 13·31-s − 32-s − 34-s + 8·37-s + 16·38-s + 12·41-s − 20·43-s + 36·44-s − 4·46-s − 15·47-s + 59·49-s + 42·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3/2·4-s − 4.53·7-s − 1.06·8-s − 3.61·11-s − 3.88·13-s − 3.20·14-s + 16-s − 0.242·17-s + 3.67·19-s − 2.55·22-s − 0.834·23-s − 2.74·26-s + 6.80·28-s − 0.371·29-s + 2.33·31-s − 0.176·32-s − 0.171·34-s + 1.31·37-s + 2.59·38-s + 1.87·41-s − 3.04·43-s + 5.42·44-s − 0.589·46-s − 2.18·47-s + 59/7·49-s + 5.82·52-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.65652\times 10^{13}\) |
| Root analytic conductor: | \(6.70192\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{16} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| good | 2 | \( 1 - T + p^{2} T^{2} - p^{2} T^{3} + 9 T^{4} - p^{2} T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} - p^{5} T^{11} + 9 p^{4} T^{12} - p^{7} T^{13} + p^{8} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 12 T + 85 T^{2} + 446 T^{3} + 1936 T^{4} + 7218 T^{5} + 23861 T^{6} + 71440 T^{7} + 196779 T^{8} + 71440 p T^{9} + 23861 p^{2} T^{10} + 7218 p^{3} T^{11} + 1936 p^{4} T^{12} + 446 p^{5} T^{13} + 85 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 12 T + 90 T^{2} + 469 T^{3} + 161 p T^{4} + 4307 T^{5} + 2938 T^{6} - 30154 T^{7} - 153632 T^{8} - 30154 p T^{9} + 2938 p^{2} T^{10} + 4307 p^{3} T^{11} + 161 p^{5} T^{12} + 469 p^{5} T^{13} + 90 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 14 T + 147 T^{2} + 1064 T^{3} + 6722 T^{4} + 35176 T^{5} + 999 p^{2} T^{6} + 704458 T^{7} + 2715097 T^{8} + 704458 p T^{9} + 999 p^{4} T^{10} + 35176 p^{3} T^{11} + 6722 p^{4} T^{12} + 1064 p^{5} T^{13} + 147 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + T + 48 T^{2} - 3 T^{3} + 1201 T^{4} - 1739 T^{5} + 21400 T^{6} - 62259 T^{7} + 346684 T^{8} - 62259 p T^{9} + 21400 p^{2} T^{10} - 1739 p^{3} T^{11} + 1201 p^{4} T^{12} - 3 p^{5} T^{13} + 48 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 16 T + 203 T^{2} - 1574 T^{3} + 10318 T^{4} - 47046 T^{5} + 192025 T^{6} - 567810 T^{7} + 2356491 T^{8} - 567810 p T^{9} + 192025 p^{2} T^{10} - 47046 p^{3} T^{11} + 10318 p^{4} T^{12} - 1574 p^{5} T^{13} + 203 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 4 T + 80 T^{2} + 218 T^{3} + 3191 T^{4} + 5396 T^{5} + 91104 T^{6} + 110830 T^{7} + 2230464 T^{8} + 110830 p T^{9} + 91104 p^{2} T^{10} + 5396 p^{3} T^{11} + 3191 p^{4} T^{12} + 218 p^{5} T^{13} + 80 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 2 T + 106 T^{2} + 149 T^{3} + 5355 T^{4} + 1367 T^{5} + 179150 T^{6} - 179890 T^{7} + 5197016 T^{8} - 179890 p T^{9} + 179150 p^{2} T^{10} + 1367 p^{3} T^{11} + 5355 p^{4} T^{12} + 149 p^{5} T^{13} + 106 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 13 T + 152 T^{2} - 1368 T^{3} + 11853 T^{4} - 85563 T^{5} + 582220 T^{6} - 3607325 T^{7} + 21326621 T^{8} - 3607325 p T^{9} + 582220 p^{2} T^{10} - 85563 p^{3} T^{11} + 11853 p^{4} T^{12} - 1368 p^{5} T^{13} + 152 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T + 265 T^{2} - 1759 T^{3} + 30976 T^{4} - 172912 T^{5} + 2132366 T^{6} - 9991070 T^{7} + 96080589 T^{8} - 9991070 p T^{9} + 2132366 p^{2} T^{10} - 172912 p^{3} T^{11} + 30976 p^{4} T^{12} - 1759 p^{5} T^{13} + 265 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 12 T + 322 T^{2} - 2957 T^{3} + 44843 T^{4} - 330527 T^{5} + 3598030 T^{6} - 21557760 T^{7} + 182961816 T^{8} - 21557760 p T^{9} + 3598030 p^{2} T^{10} - 330527 p^{3} T^{11} + 44843 p^{4} T^{12} - 2957 p^{5} T^{13} + 322 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 20 T + 350 T^{2} + 4400 T^{3} + 49591 T^{4} + 467330 T^{5} + 4034125 T^{6} + 30445275 T^{7} + 212290161 T^{8} + 30445275 p T^{9} + 4034125 p^{2} T^{10} + 467330 p^{3} T^{11} + 49591 p^{4} T^{12} + 4400 p^{5} T^{13} + 350 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 15 T + 305 T^{2} + 3300 T^{3} + 41661 T^{4} + 360570 T^{5} + 3486535 T^{6} + 25020645 T^{7} + 196992836 T^{8} + 25020645 p T^{9} + 3486535 p^{2} T^{10} + 360570 p^{3} T^{11} + 41661 p^{4} T^{12} + 3300 p^{5} T^{13} + 305 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 4 T + 5 p T^{2} + 1178 T^{3} + 35401 T^{4} + 155116 T^{5} + 3127779 T^{6} + 12258590 T^{7} + 195814324 T^{8} + 12258590 p T^{9} + 3127779 p^{2} T^{10} + 155116 p^{3} T^{11} + 35401 p^{4} T^{12} + 1178 p^{5} T^{13} + 5 p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 14 T + 308 T^{2} + 3541 T^{3} + 46073 T^{4} + 424549 T^{5} + 4330440 T^{6} + 33750410 T^{7} + 292522356 T^{8} + 33750410 p T^{9} + 4330440 p^{2} T^{10} + 424549 p^{3} T^{11} + 46073 p^{4} T^{12} + 3541 p^{5} T^{13} + 308 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 10 T + 302 T^{2} - 2210 T^{3} + 40633 T^{4} - 237420 T^{5} + 3532399 T^{6} - 17630955 T^{7} + 236872815 T^{8} - 17630955 p T^{9} + 3532399 p^{2} T^{10} - 237420 p^{3} T^{11} + 40633 p^{4} T^{12} - 2210 p^{5} T^{13} + 302 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 19 T + 353 T^{2} + 4628 T^{3} + 63916 T^{4} + 671229 T^{5} + 7103470 T^{6} + 63200124 T^{7} + 570267789 T^{8} + 63200124 p T^{9} + 7103470 p^{2} T^{10} + 671229 p^{3} T^{11} + 63916 p^{4} T^{12} + 4628 p^{5} T^{13} + 353 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 21 T + 550 T^{2} + 8335 T^{3} + 131315 T^{4} + 1549183 T^{5} + 18114598 T^{6} + 171684165 T^{7} + 1591301520 T^{8} + 171684165 p T^{9} + 18114598 p^{2} T^{10} + 1549183 p^{3} T^{11} + 131315 p^{4} T^{12} + 8335 p^{5} T^{13} + 550 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 19 T + 605 T^{2} - 8458 T^{3} + 154546 T^{4} - 1696171 T^{5} + 22411474 T^{6} - 197652450 T^{7} + 2038458369 T^{8} - 197652450 p T^{9} + 22411474 p^{2} T^{10} - 1696171 p^{3} T^{11} + 154546 p^{4} T^{12} - 8458 p^{5} T^{13} + 605 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 10 T + 312 T^{2} - 3010 T^{3} + 51533 T^{4} - 437210 T^{5} + 5828124 T^{6} - 42525800 T^{7} + 514315885 T^{8} - 42525800 p T^{9} + 5828124 p^{2} T^{10} - 437210 p^{3} T^{11} + 51533 p^{4} T^{12} - 3010 p^{5} T^{13} + 312 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 27 T + 796 T^{2} + 13847 T^{3} + 239279 T^{4} + 3100853 T^{5} + 39301012 T^{6} + 403360281 T^{7} + 4042436448 T^{8} + 403360281 p T^{9} + 39301012 p^{2} T^{10} + 3100853 p^{3} T^{11} + 239279 p^{4} T^{12} + 13847 p^{5} T^{13} + 796 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 9 T + 418 T^{2} - 2821 T^{3} + 87473 T^{4} - 507769 T^{5} + 12418210 T^{6} - 62255505 T^{7} + 1276458036 T^{8} - 62255505 p T^{9} + 12418210 p^{2} T^{10} - 507769 p^{3} T^{11} + 87473 p^{4} T^{12} - 2821 p^{5} T^{13} + 418 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 24 T + 863 T^{2} + 14828 T^{3} + 304496 T^{4} + 4040944 T^{5} + 59789185 T^{6} + 631650504 T^{7} + 7266909259 T^{8} + 631650504 p T^{9} + 59789185 p^{2} T^{10} + 4040944 p^{3} T^{11} + 304496 p^{4} T^{12} + 14828 p^{5} T^{13} + 863 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.82335772724900092353041272270, −3.40797128737363002852747865029, −3.36293025179788595039817564330, −3.34384667476243493924791062447, −3.34050986367958883944484588318, −3.23458343891157703440628068121, −3.08167016805999526600974950944, −2.95728519280046370836826036429, −2.86262142231049834533115228714, −2.82647771875711633003213684109, −2.74096162697275333777004906385, −2.68188843752185624651541641438, −2.61663723831120681166400190294, −2.39621231298825841864660345969, −2.30136488946697099526484441480, −2.28988264519204670024297175577, −1.99888206222856162963453314670, −1.95633770104611575952311389096, −1.84320793015606911521114363983, −1.30561839119548029125384698753, −1.25967463027440708238426845823, −1.22354910764416862564520334922, −1.04985205671398068911079983900, −1.03322072827411875373184260472, −0.928550613724677606401776795234, 0, 0, 0, 0, 0, 0, 0, 0, 0.928550613724677606401776795234, 1.03322072827411875373184260472, 1.04985205671398068911079983900, 1.22354910764416862564520334922, 1.25967463027440708238426845823, 1.30561839119548029125384698753, 1.84320793015606911521114363983, 1.95633770104611575952311389096, 1.99888206222856162963453314670, 2.28988264519204670024297175577, 2.30136488946697099526484441480, 2.39621231298825841864660345969, 2.61663723831120681166400190294, 2.68188843752185624651541641438, 2.74096162697275333777004906385, 2.82647771875711633003213684109, 2.86262142231049834533115228714, 2.95728519280046370836826036429, 3.08167016805999526600974950944, 3.23458343891157703440628068121, 3.34050986367958883944484588318, 3.34384667476243493924791062447, 3.36293025179788595039817564330, 3.40797128737363002852747865029, 3.82335772724900092353041272270
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.