Dirichlet series
| L(s) = 1 | + 4·2-s + 8·4-s − 6·5-s + 10·7-s + 16·8-s − 6·9-s − 24·10-s + 24·11-s + 40·14-s + 36·16-s − 84·17-s − 24·18-s + 10·19-s − 48·20-s + 96·22-s − 12·23-s + 18·25-s + 80·28-s + 36·29-s − 94·31-s + 64·32-s − 336·34-s − 60·35-s − 48·36-s + 140·37-s + 40·38-s − 96·40-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·4-s − 6/5·5-s + 10/7·7-s + 2·8-s − 2/3·9-s − 2.39·10-s + 2.18·11-s + 20/7·14-s + 9/4·16-s − 4.94·17-s − 4/3·18-s + 0.526·19-s − 2.39·20-s + 4.36·22-s − 0.521·23-s + 0.719·25-s + 20/7·28-s + 1.24·29-s − 3.03·31-s + 2·32-s − 9.88·34-s − 1.71·35-s − 4/3·36-s + 3.78·37-s + 1.05·38-s − 2.39·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(416.329\) |
| Root analytic conductor: | \(1.45785\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 13^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(6.838218798\) |
| \(L(\frac12)\) | \(\approx\) | \(6.838218798\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | |
| 13 | \( 1 - 50 T^{2} - 35181 T^{4} - 50 p^{4} T^{6} + p^{8} T^{8} \) | |
| good | 5 | \( 1 + 6 T + 18 T^{2} - 132 T^{3} - 547 p T^{4} - 11484 T^{5} - 10962 T^{6} + 279258 T^{7} + 2657664 T^{8} + 279258 p^{2} T^{9} - 10962 p^{4} T^{10} - 11484 p^{6} T^{11} - 547 p^{9} T^{12} - 132 p^{10} T^{13} + 18 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( 1 - 10 T + 29 T^{2} - 2 p T^{3} + 2183 T^{4} - 596 p T^{5} - 112608 T^{6} + 1269948 T^{7} - 6907766 T^{8} + 1269948 p^{2} T^{9} - 112608 p^{4} T^{10} - 596 p^{7} T^{11} + 2183 p^{8} T^{12} - 2 p^{11} T^{13} + 29 p^{12} T^{14} - 10 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 - 24 T + 300 T^{2} - 300 p T^{3} + 49870 T^{4} - 522396 T^{5} + 3395280 T^{6} - 30534804 T^{7} + 430499811 T^{8} - 30534804 p^{2} T^{9} + 3395280 p^{4} T^{10} - 522396 p^{6} T^{11} + 49870 p^{8} T^{12} - 300 p^{11} T^{13} + 300 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 + 84 T + 239 p T^{2} + 143724 T^{3} + 4034965 T^{4} + 94551624 T^{5} + 114148742 p T^{6} + 36125515344 T^{7} + 629895993598 T^{8} + 36125515344 p^{2} T^{9} + 114148742 p^{5} T^{10} + 94551624 p^{6} T^{11} + 4034965 p^{8} T^{12} + 143724 p^{10} T^{13} + 239 p^{13} T^{14} + 84 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 10 T + 842 T^{2} - 5432 T^{3} + 254894 T^{4} - 4601174 T^{5} + 62962224 T^{6} - 195378966 p T^{7} + 23878908259 T^{8} - 195378966 p^{3} T^{9} + 62962224 p^{4} T^{10} - 4601174 p^{6} T^{11} + 254894 p^{8} T^{12} - 5432 p^{10} T^{13} + 842 p^{12} T^{14} - 10 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 + 12 T + 808 T^{2} + 9120 T^{3} + 15566 p T^{4} - 6278580 T^{5} - 37339520 T^{6} - 7341809748 T^{7} - 111601535981 T^{8} - 7341809748 p^{2} T^{9} - 37339520 p^{4} T^{10} - 6278580 p^{6} T^{11} + 15566 p^{9} T^{12} + 9120 p^{10} T^{13} + 808 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 36 T - 1081 T^{2} + 37668 T^{3} + 627805 T^{4} + 3595128 T^{5} - 1371286354 T^{6} - 8356031088 T^{7} + 1714447657438 T^{8} - 8356031088 p^{2} T^{9} - 1371286354 p^{4} T^{10} + 3595128 p^{6} T^{11} + 627805 p^{8} T^{12} + 37668 p^{10} T^{13} - 1081 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 + 94 T + 4418 T^{2} + 169400 T^{3} + 6442529 T^{4} + 230690348 T^{5} + 7877190 p^{2} T^{6} + 7537605690 p T^{7} + 7138777033984 T^{8} + 7537605690 p^{3} T^{9} + 7877190 p^{6} T^{10} + 230690348 p^{6} T^{11} + 6442529 p^{8} T^{12} + 169400 p^{10} T^{13} + 4418 p^{12} T^{14} + 94 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 - 140 T + 8867 T^{2} - 331828 T^{3} + 7978181 T^{4} - 18496720 T^{5} - 13571631882 T^{6} + 996881920368 T^{7} - 44216538884138 T^{8} + 996881920368 p^{2} T^{9} - 13571631882 p^{4} T^{10} - 18496720 p^{6} T^{11} + 7978181 p^{8} T^{12} - 331828 p^{10} T^{13} + 8867 p^{12} T^{14} - 140 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 72 T + 5217 T^{2} - 190188 T^{3} + 8440915 T^{4} - 135489804 T^{5} + 3556160880 T^{6} + 285967488756 T^{7} - 9392296265346 T^{8} + 285967488756 p^{2} T^{9} + 3556160880 p^{4} T^{10} - 135489804 p^{6} T^{11} + 8440915 p^{8} T^{12} - 190188 p^{10} T^{13} + 5217 p^{12} T^{14} - 72 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 222 T + 28459 T^{2} + 2670882 T^{3} + 200826889 T^{4} + 295624188 p T^{5} + 703243755934 T^{6} + 809068195320 p T^{7} + 1564414661688946 T^{8} + 809068195320 p^{3} T^{9} + 703243755934 p^{4} T^{10} + 295624188 p^{7} T^{11} + 200826889 p^{8} T^{12} + 2670882 p^{10} T^{13} + 28459 p^{12} T^{14} + 222 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 - 300 T + 45000 T^{2} - 4678428 T^{3} + 389267620 T^{4} - 27657860724 T^{5} + 1724159592792 T^{6} - 95617768295268 T^{7} + 4746944876916294 T^{8} - 95617768295268 p^{2} T^{9} + 1724159592792 p^{4} T^{10} - 27657860724 p^{6} T^{11} + 389267620 p^{8} T^{12} - 4678428 p^{10} T^{13} + 45000 p^{12} T^{14} - 300 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( ( 1 - 42 T + 9025 T^{2} - 319362 T^{3} + 34812360 T^{4} - 319362 p^{2} T^{5} + 9025 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 59 | \( 1 + 60 T + 5892 T^{2} + 206124 T^{3} + 13736014 T^{4} - 362622000 T^{5} - 19547251680 T^{6} - 5972694061296 T^{7} - 299130704667789 T^{8} - 5972694061296 p^{2} T^{9} - 19547251680 p^{4} T^{10} - 362622000 p^{6} T^{11} + 13736014 p^{8} T^{12} + 206124 p^{10} T^{13} + 5892 p^{12} T^{14} + 60 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 + 90 T - 1102 T^{2} + 301236 T^{3} + 37355365 T^{4} - 844763964 T^{5} + 18752573750 T^{6} + 4281081079470 T^{7} - 230418365948828 T^{8} + 4281081079470 p^{2} T^{9} + 18752573750 p^{4} T^{10} - 844763964 p^{6} T^{11} + 37355365 p^{8} T^{12} + 301236 p^{10} T^{13} - 1102 p^{12} T^{14} + 90 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 304 T + 39965 T^{2} - 2820476 T^{3} + 98159603 T^{4} - 2033824556 T^{5} + 506161675704 T^{6} - 87621939622044 T^{7} + 7722813169084774 T^{8} - 87621939622044 p^{2} T^{9} + 506161675704 p^{4} T^{10} - 2033824556 p^{6} T^{11} + 98159603 p^{8} T^{12} - 2820476 p^{10} T^{13} + 39965 p^{12} T^{14} - 304 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 + 192 T + 10092 T^{2} - 555084 T^{3} - 101576786 T^{4} - 5546929596 T^{5} - 45185376912 T^{6} + 17715378667788 T^{7} + 1777490142285891 T^{8} + 17715378667788 p^{2} T^{9} - 45185376912 p^{4} T^{10} - 5546929596 p^{6} T^{11} - 101576786 p^{8} T^{12} - 555084 p^{10} T^{13} + 10092 p^{12} T^{14} + 192 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 - 16 T + 128 T^{2} + 140416 T^{3} + 32181122 T^{4} - 2514741392 T^{5} + 45975005184 T^{6} - 6743944510608 T^{7} + 38152958606083 T^{8} - 6743944510608 p^{2} T^{9} + 45975005184 p^{4} T^{10} - 2514741392 p^{6} T^{11} + 32181122 p^{8} T^{12} + 140416 p^{10} T^{13} + 128 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( ( 1 + 48 T + 13297 T^{2} + 123072 T^{3} + 79863648 T^{4} + 123072 p^{2} T^{5} + 13297 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 83 | \( 1 + 682176 T^{3} - 7433372 T^{4} - 4619013696 T^{5} + 232682047488 T^{6} - 1985641063296 T^{7} - 3794632805852922 T^{8} - 1985641063296 p^{2} T^{9} + 232682047488 p^{4} T^{10} - 4619013696 p^{6} T^{11} - 7433372 p^{8} T^{12} + 682176 p^{10} T^{13} + p^{16} T^{16} \) | |
| 89 | \( 1 - 354 T + 89994 T^{2} - 16520520 T^{3} + 2570396254 T^{4} - 336942863190 T^{5} + 39369792597600 T^{6} - 4070957803409466 T^{7} + 382780797228102051 T^{8} - 4070957803409466 p^{2} T^{9} + 39369792597600 p^{4} T^{10} - 336942863190 p^{6} T^{11} + 2570396254 p^{8} T^{12} - 16520520 p^{10} T^{13} + 89994 p^{12} T^{14} - 354 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 + 460 T + 108755 T^{2} + 17098460 T^{3} + 1992314285 T^{4} + 183186207152 T^{5} + 13977188124270 T^{6} + 972489780066792 T^{7} + 78525217013648422 T^{8} + 972489780066792 p^{2} T^{9} + 13977188124270 p^{4} T^{10} + 183186207152 p^{6} T^{11} + 1992314285 p^{8} T^{12} + 17098460 p^{10} T^{13} + 108755 p^{12} T^{14} + 460 p^{14} T^{15} + p^{16} T^{16} \) | |
| 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
−6.55011400135775077158026706076, −6.41204933831995675619945779618, −6.11075728865663418035968654741, −6.08742647797107526928395949237, −5.89714304195550361863896705593, −5.58224358065505962444410043928, −5.48194545019363236179987928244, −5.11250641520283181732055064418, −5.01986733385761207909467476060, −4.99907588882584987091790229373, −4.52559726024657033911490971382, −4.23739620863164218609186001809, −4.23532752786441553675729096662, −4.17255629601728924615184353100, −4.07782171265526942671399339260, −3.93440274553145140442271240797, −3.75848604698402290349650874034, −3.18526005980408196591463513336, −2.90507961410787414679364182459, −2.60663425455804175641434139017, −2.26833808309964379848734156608, −2.26404979982670582837484680998, −1.71825785398380507253526867395, −1.34172481331503456514989208179, −0.58637852393815146177077860459, 0.58637852393815146177077860459, 1.34172481331503456514989208179, 1.71825785398380507253526867395, 2.26404979982670582837484680998, 2.26833808309964379848734156608, 2.60663425455804175641434139017, 2.90507961410787414679364182459, 3.18526005980408196591463513336, 3.75848604698402290349650874034, 3.93440274553145140442271240797, 4.07782171265526942671399339260, 4.17255629601728924615184353100, 4.23532752786441553675729096662, 4.23739620863164218609186001809, 4.52559726024657033911490971382, 4.99907588882584987091790229373, 5.01986733385761207909467476060, 5.11250641520283181732055064418, 5.48194545019363236179987928244, 5.58224358065505962444410043928, 5.89714304195550361863896705593, 6.08742647797107526928395949237, 6.11075728865663418035968654741, 6.41204933831995675619945779618, 6.55011400135775077158026706076
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.