Dirichlet series
| L(s) = 1 | + 8·2-s − 8·3-s + 36·4-s − 4·5-s − 64·6-s + 120·8-s + 36·9-s − 32·10-s + 8·11-s − 288·12-s + 4·13-s + 32·15-s + 330·16-s − 8·17-s + 288·18-s − 8·19-s − 144·20-s + 64·22-s + 4·23-s − 960·24-s + 32·26-s − 120·27-s + 16·29-s + 256·30-s − 4·31-s + 792·32-s − 64·33-s + ⋯ |
| L(s) = 1 | + 5.65·2-s − 4.61·3-s + 18·4-s − 1.78·5-s − 26.1·6-s + 42.4·8-s + 12·9-s − 10.1·10-s + 2.41·11-s − 83.1·12-s + 1.10·13-s + 8.26·15-s + 82.5·16-s − 1.94·17-s + 67.8·18-s − 1.83·19-s − 32.1·20-s + 13.6·22-s + 0.834·23-s − 195.·24-s + 6.27·26-s − 23.0·27-s + 2.97·29-s + 46.7·30-s − 0.718·31-s + 140.·32-s − 11.1·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 7^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.56683\times 10^{13}\) |
| Root analytic conductor: | \(6.67865\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(473.1033693\) |
| \(L(\frac12)\) | \(\approx\) | \(473.1033693\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( ( 1 + T )^{8} \) | |
| 7 | \( 1 \) | |
| 19 | \( ( 1 + T )^{8} \) | |
| good | 5 | \( 1 + 4 T + 16 T^{2} + 36 T^{3} + 102 T^{4} + 204 T^{5} + 568 T^{6} + 1292 T^{7} + 3442 T^{8} + 1292 p T^{9} + 568 p^{2} T^{10} + 204 p^{3} T^{11} + 102 p^{4} T^{12} + 36 p^{5} T^{13} + 16 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 8 T + 60 T^{2} - 248 T^{3} + 1008 T^{4} - 232 p T^{5} + 7892 T^{6} - 14472 T^{7} + 60574 T^{8} - 14472 p T^{9} + 7892 p^{2} T^{10} - 232 p^{4} T^{11} + 1008 p^{4} T^{12} - 248 p^{5} T^{13} + 60 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 4 T + 24 T^{2} - 92 T^{3} + 566 T^{4} - 1444 T^{5} + 560 p T^{6} - 19548 T^{7} + 105554 T^{8} - 19548 p T^{9} + 560 p^{3} T^{10} - 1444 p^{3} T^{11} + 566 p^{4} T^{12} - 92 p^{5} T^{13} + 24 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 8 T + 88 T^{2} + 472 T^{3} + 3220 T^{4} + 13064 T^{5} + 70600 T^{6} + 241432 T^{7} + 1237142 T^{8} + 241432 p T^{9} + 70600 p^{2} T^{10} + 13064 p^{3} T^{11} + 3220 p^{4} T^{12} + 472 p^{5} T^{13} + 88 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 4 T + 132 T^{2} - 564 T^{3} + 8386 T^{4} - 34228 T^{5} + 338012 T^{6} - 1210660 T^{7} + 9328562 T^{8} - 1210660 p T^{9} + 338012 p^{2} T^{10} - 34228 p^{3} T^{11} + 8386 p^{4} T^{12} - 564 p^{5} T^{13} + 132 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 16 T + 192 T^{2} - 1520 T^{3} + 11004 T^{4} - 68880 T^{5} + 449728 T^{6} - 2684784 T^{7} + 15558054 T^{8} - 2684784 p T^{9} + 449728 p^{2} T^{10} - 68880 p^{3} T^{11} + 11004 p^{4} T^{12} - 1520 p^{5} T^{13} + 192 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 4 T + 120 T^{2} + 252 T^{3} + 7134 T^{4} + 10316 T^{5} + 329808 T^{6} + 453204 T^{7} + 11889810 T^{8} + 453204 p T^{9} + 329808 p^{2} T^{10} + 10316 p^{3} T^{11} + 7134 p^{4} T^{12} + 252 p^{5} T^{13} + 120 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 12 T + 184 T^{2} - 1308 T^{3} + 11646 T^{4} - 48196 T^{5} + 305184 T^{6} - 383892 T^{7} + 5698706 T^{8} - 383892 p T^{9} + 305184 p^{2} T^{10} - 48196 p^{3} T^{11} + 11646 p^{4} T^{12} - 1308 p^{5} T^{13} + 184 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 100 T^{2} + 32 T^{3} + 6952 T^{4} - 864 T^{5} + 405228 T^{6} + 53312 T^{7} + 18680462 T^{8} + 53312 p T^{9} + 405228 p^{2} T^{10} - 864 p^{3} T^{11} + 6952 p^{4} T^{12} + 32 p^{5} T^{13} + 100 p^{6} T^{14} + p^{8} T^{16} \) | |
| 43 | \( 1 - 16 T + 140 T^{2} - 336 T^{3} - 2008 T^{4} + 19504 T^{5} + 87620 T^{6} - 1857040 T^{7} + 16829806 T^{8} - 1857040 p T^{9} + 87620 p^{2} T^{10} + 19504 p^{3} T^{11} - 2008 p^{4} T^{12} - 336 p^{5} T^{13} + 140 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 12 T + 176 T^{2} + 2012 T^{3} + 20302 T^{4} + 179772 T^{5} + 1501048 T^{6} + 11411948 T^{7} + 81086962 T^{8} + 11411948 p T^{9} + 1501048 p^{2} T^{10} + 179772 p^{3} T^{11} + 20302 p^{4} T^{12} + 2012 p^{5} T^{13} + 176 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 24 T + 376 T^{2} - 4008 T^{3} + 34012 T^{4} - 198040 T^{5} + 745352 T^{6} + 567320 T^{7} - 18040026 T^{8} + 567320 p T^{9} + 745352 p^{2} T^{10} - 198040 p^{3} T^{11} + 34012 p^{4} T^{12} - 4008 p^{5} T^{13} + 376 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 8 T + 284 T^{2} - 2488 T^{3} + 40872 T^{4} - 361112 T^{5} + 3895156 T^{6} - 32122600 T^{7} + 267758190 T^{8} - 32122600 p T^{9} + 3895156 p^{2} T^{10} - 361112 p^{3} T^{11} + 40872 p^{4} T^{12} - 2488 p^{5} T^{13} + 284 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 252 T^{2} + 640 T^{3} + 34312 T^{4} + 111808 T^{5} + 3357908 T^{6} + 10424512 T^{7} + 238233710 T^{8} + 10424512 p T^{9} + 3357908 p^{2} T^{10} + 111808 p^{3} T^{11} + 34312 p^{4} T^{12} + 640 p^{5} T^{13} + 252 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 - 8 T + 332 T^{2} - 1464 T^{3} + 41008 T^{4} - 23992 T^{5} + 2477060 T^{6} + 11328632 T^{7} + 122773470 T^{8} + 11328632 p T^{9} + 2477060 p^{2} T^{10} - 23992 p^{3} T^{11} + 41008 p^{4} T^{12} - 1464 p^{5} T^{13} + 332 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 24 T + 536 T^{2} - 7768 T^{3} + 104508 T^{4} - 1102776 T^{5} + 11296424 T^{6} - 98900216 T^{7} + 883314118 T^{8} - 98900216 p T^{9} + 11296424 p^{2} T^{10} - 1102776 p^{3} T^{11} + 104508 p^{4} T^{12} - 7768 p^{5} T^{13} + 536 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 16 T + 356 T^{2} - 5040 T^{3} + 67720 T^{4} - 792144 T^{5} + 8479212 T^{6} - 81026544 T^{7} + 739911886 T^{8} - 81026544 p T^{9} + 8479212 p^{2} T^{10} - 792144 p^{3} T^{11} + 67720 p^{4} T^{12} - 5040 p^{5} T^{13} + 356 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 20 T + 532 T^{2} - 7236 T^{3} + 113522 T^{4} - 1197764 T^{5} + 14301228 T^{6} - 126892212 T^{7} + 1288288786 T^{8} - 126892212 p T^{9} + 14301228 p^{2} T^{10} - 1197764 p^{3} T^{11} + 113522 p^{4} T^{12} - 7236 p^{5} T^{13} + 532 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 16 T + 488 T^{2} + 5200 T^{3} + 94172 T^{4} + 729552 T^{5} + 10724504 T^{6} + 66887696 T^{7} + 944938470 T^{8} + 66887696 p T^{9} + 10724504 p^{2} T^{10} + 729552 p^{3} T^{11} + 94172 p^{4} T^{12} + 5200 p^{5} T^{13} + 488 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 272 T^{2} + 544 T^{3} + 26916 T^{4} + 108320 T^{5} + 1173040 T^{6} + 7959104 T^{7} + 40601606 T^{8} + 7959104 p T^{9} + 1173040 p^{2} T^{10} + 108320 p^{3} T^{11} + 26916 p^{4} T^{12} + 544 p^{5} T^{13} + 272 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 - 8 T + 424 T^{2} - 4408 T^{3} + 92084 T^{4} - 1064296 T^{5} + 14009720 T^{6} - 153773144 T^{7} + 1581892310 T^{8} - 153773144 p T^{9} + 14009720 p^{2} T^{10} - 1064296 p^{3} T^{11} + 92084 p^{4} T^{12} - 4408 p^{5} T^{13} + 424 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.69705062622968668316729108699, −3.34371653406661416881212603134, −3.19197266347394780745939485577, −2.98809100483567058651483445719, −2.89009355460509420352179645553, −2.87293347465536735259532927647, −2.86313301777521331499930748331, −2.75115161154838930910488409880, −2.74393601613416501002927411464, −2.16139919142620796768019817315, −2.05901699312838220533081007540, −2.05685403467351135380633816750, −1.92375937773845908669945090819, −1.85726818912877732063659177921, −1.84576472733671239671714685536, −1.80689733939929103081540324152, −1.76575581703738971465047641933, −1.13021298474506601424669403145, −1.06582849682692811140751420909, −0.888020471345344422688321449838, −0.790437486101857919170642129070, −0.77696660038852330870514129752, −0.70272824203797912375097795882, −0.51415756832339118356260165421, −0.38012907181428562193576513315, 0.38012907181428562193576513315, 0.51415756832339118356260165421, 0.70272824203797912375097795882, 0.77696660038852330870514129752, 0.790437486101857919170642129070, 0.888020471345344422688321449838, 1.06582849682692811140751420909, 1.13021298474506601424669403145, 1.76575581703738971465047641933, 1.80689733939929103081540324152, 1.84576472733671239671714685536, 1.85726818912877732063659177921, 1.92375937773845908669945090819, 2.05685403467351135380633816750, 2.05901699312838220533081007540, 2.16139919142620796768019817315, 2.74393601613416501002927411464, 2.75115161154838930910488409880, 2.86313301777521331499930748331, 2.87293347465536735259532927647, 2.89009355460509420352179645553, 2.98809100483567058651483445719, 3.19197266347394780745939485577, 3.34371653406661416881212603134, 3.69705062622968668316729108699
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.