Dirichlet series
| L(s) = 1 | + 5·2-s + 5·3-s + 10·4-s + 25·6-s + 5·8-s + 10·9-s − 4·11-s + 50·12-s + 5·13-s − 16·16-s + 10·17-s + 50·18-s − 5·19-s − 20·22-s − 5·23-s + 25·24-s + 25·26-s + 5·27-s − 5·29-s − 9·31-s − 30·32-s − 20·33-s + 50·34-s + 100·36-s − 30·37-s − 25·38-s + 25·39-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 2.88·3-s + 5·4-s + 10.2·6-s + 1.76·8-s + 10/3·9-s − 1.20·11-s + 14.4·12-s + 1.38·13-s − 4·16-s + 2.42·17-s + 11.7·18-s − 1.14·19-s − 4.26·22-s − 1.04·23-s + 5.10·24-s + 4.90·26-s + 0.962·27-s − 0.928·29-s − 1.61·31-s − 5.30·32-s − 3.48·33-s + 8.57·34-s + 50/3·36-s − 4.93·37-s − 4.05·38-s + 4.00·39-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.985137\) |
| Root analytic conductor: | \(0.999064\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(12.27750680\) |
| \(L(\frac12)\) | \(\approx\) | \(12.27750680\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 5 T + 15 T^{2} - 15 p T^{3} + 41 T^{4} - 15 p T^{5} - 5 p^{2} T^{6} + 55 p T^{7} - 199 T^{8} + 55 p^{2} T^{9} - 5 p^{4} T^{10} - 15 p^{4} T^{11} + 41 p^{4} T^{12} - 15 p^{6} T^{13} + 15 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - 5 T + 5 p T^{2} - 10 p T^{3} + 4 p^{2} T^{4} - 5 T^{5} - 40 p T^{6} + 400 T^{7} - 809 T^{8} + 400 p T^{9} - 40 p^{3} T^{10} - 5 p^{3} T^{11} + 4 p^{6} T^{12} - 10 p^{6} T^{13} + 5 p^{7} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 5 p T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 7045 p^{2} T^{10} + 611 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \) | |
| 11 | \( ( 1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 36 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 - 5 T + 30 T^{2} - 60 T^{3} + 346 T^{4} - 655 T^{5} + 6335 T^{6} - 13320 T^{7} + 88856 T^{8} - 13320 p T^{9} + 6335 p^{2} T^{10} - 655 p^{3} T^{11} + 346 p^{4} T^{12} - 60 p^{5} T^{13} + 30 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 10 T + 90 T^{2} - 720 T^{3} + 4451 T^{4} - 26310 T^{5} + 136780 T^{6} - 638720 T^{7} + 2802941 T^{8} - 638720 p T^{9} + 136780 p^{2} T^{10} - 26310 p^{3} T^{11} + 4451 p^{4} T^{12} - 720 p^{5} T^{13} + 90 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 5 T - 8 T^{2} + 40 T^{3} + 878 T^{4} + 1705 T^{5} - 1861 T^{6} + 31550 T^{7} + 293380 T^{8} + 31550 p T^{9} - 1861 p^{2} T^{10} + 1705 p^{3} T^{11} + 878 p^{4} T^{12} + 40 p^{5} T^{13} - 8 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 5 T + 45 T^{2} - 100 T^{3} - 104 T^{4} - 7645 T^{5} + 18890 T^{6} - 8420 T^{7} + 1238691 T^{8} - 8420 p T^{9} + 18890 p^{2} T^{10} - 7645 p^{3} T^{11} - 104 p^{4} T^{12} - 100 p^{5} T^{13} + 45 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 5 T - 28 T^{2} + 140 T^{3} + 1268 T^{4} - 5045 T^{5} + 48219 T^{6} + 207000 T^{7} - 1738520 T^{8} + 207000 p T^{9} + 48219 p^{2} T^{10} - 5045 p^{3} T^{11} + 1268 p^{4} T^{12} + 140 p^{5} T^{13} - 28 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 9 T + 55 T^{2} + 390 T^{3} + 2980 T^{4} + 20297 T^{5} + 114748 T^{6} + 589990 T^{7} + 3582095 T^{8} + 589990 p T^{9} + 114748 p^{2} T^{10} + 20297 p^{3} T^{11} + 2980 p^{4} T^{12} + 390 p^{5} T^{13} + 55 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 30 T + 480 T^{2} + 5675 T^{3} + 56171 T^{4} + 489630 T^{5} + 3826535 T^{6} + 26904445 T^{7} + 171416106 T^{8} + 26904445 p T^{9} + 3826535 p^{2} T^{10} + 489630 p^{3} T^{11} + 56171 p^{4} T^{12} + 5675 p^{5} T^{13} + 480 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 4 T - 30 T^{2} - 240 T^{3} + 195 T^{4} + 18372 T^{5} + 77388 T^{6} - 230240 T^{7} - 1438395 T^{8} - 230240 p T^{9} + 77388 p^{2} T^{10} + 18372 p^{3} T^{11} + 195 p^{4} T^{12} - 240 p^{5} T^{13} - 30 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 5 p T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 1578205 p^{2} T^{10} + 22911 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 + 110 T^{2} + 90 T^{3} + 4101 T^{4} + 9900 T^{5} - 3830 T^{6} + 910260 T^{7} - 5610889 T^{8} + 910260 p T^{9} - 3830 p^{2} T^{10} + 9900 p^{3} T^{11} + 4101 p^{4} T^{12} + 90 p^{5} T^{13} + 110 p^{6} T^{14} + p^{8} T^{16} \) | |
| 53 | \( 1 - 10 T + 100 T^{2} - 1625 T^{3} + 11531 T^{4} - 95310 T^{5} + 857875 T^{6} - 5635035 T^{7} + 42472426 T^{8} - 5635035 p T^{9} + 857875 p^{2} T^{10} - 95310 p^{3} T^{11} + 11531 p^{4} T^{12} - 1625 p^{5} T^{13} + 100 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 2 p T^{2} + 900 T^{3} + 3393 T^{4} - 96300 T^{5} + 615034 T^{6} + 2943900 T^{7} - 65890945 T^{8} + 2943900 p T^{9} + 615034 p^{2} T^{10} - 96300 p^{3} T^{11} + 3393 p^{4} T^{12} + 900 p^{5} T^{13} - 2 p^{7} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 + 9 T - 165 T^{2} - 1800 T^{3} + 7560 T^{4} + 154437 T^{5} + 526138 T^{6} - 4559670 T^{7} - 69838275 T^{8} - 4559670 p T^{9} + 526138 p^{2} T^{10} + 154437 p^{3} T^{11} + 7560 p^{4} T^{12} - 1800 p^{5} T^{13} - 165 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 20 T + 250 T^{2} + 2600 T^{3} + 26091 T^{4} + 241820 T^{5} + 2034700 T^{6} + 15361680 T^{7} + 117317461 T^{8} + 15361680 p T^{9} + 2034700 p^{2} T^{10} + 241820 p^{3} T^{11} + 26091 p^{4} T^{12} + 2600 p^{5} T^{13} + 250 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 6 T - 910 T^{3} + 4875 T^{4} + 34402 T^{5} + 474398 T^{6} - 3835500 T^{7} - 25760305 T^{8} - 3835500 p T^{9} + 474398 p^{2} T^{10} + 34402 p^{3} T^{11} + 4875 p^{4} T^{12} - 910 p^{5} T^{13} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 15 T + 195 T^{2} + 1340 T^{3} + 15786 T^{4} + 132465 T^{5} + 1900340 T^{6} + 200800 p T^{7} + 154250461 T^{8} + 200800 p^{2} T^{9} + 1900340 p^{2} T^{10} + 132465 p^{3} T^{11} + 15786 p^{4} T^{12} + 1340 p^{5} T^{13} + 195 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 15 T - 58 T^{2} + 2180 T^{3} - 7802 T^{4} - 148865 T^{5} + 1559169 T^{6} + 5584500 T^{7} - 169067020 T^{8} + 5584500 p T^{9} + 1559169 p^{2} T^{10} - 148865 p^{3} T^{11} - 7802 p^{4} T^{12} + 2180 p^{5} T^{13} - 58 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 45 T + 1115 T^{2} - 19890 T^{3} + 284376 T^{4} - 3450645 T^{5} + 448160 p T^{6} - 367888080 T^{7} + 3431240591 T^{8} - 367888080 p T^{9} + 448160 p^{3} T^{10} - 3450645 p^{3} T^{11} + 284376 p^{4} T^{12} - 19890 p^{5} T^{13} + 1115 p^{6} T^{14} - 45 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 25 T + 342 T^{2} + 5000 T^{3} + 78868 T^{4} + 928525 T^{5} + 9098049 T^{6} + 101226750 T^{7} + 1076434080 T^{8} + 101226750 p T^{9} + 9098049 p^{2} T^{10} + 928525 p^{3} T^{11} + 78868 p^{4} T^{12} + 5000 p^{5} T^{13} + 342 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 60 T + 1830 T^{2} - 35660 T^{3} + 467711 T^{4} - 3770460 T^{5} + 6583460 T^{6} + 292271720 T^{7} - 4451764059 T^{8} + 292271720 p T^{9} + 6583460 p^{2} T^{10} - 3770460 p^{3} T^{11} + 467711 p^{4} T^{12} - 35660 p^{5} T^{13} + 1830 p^{6} T^{14} - 60 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−6.15209028999264814683207120899, −5.85892989472501405165558560314, −5.83216817017281617460988907209, −5.66915721391702084461007968095, −5.52733087696146860919872411262, −5.32768155854376397828334576510, −5.15115860058602761065731726403, −5.08694410906755579888508315051, −5.00543432114936526712372385595, −4.62355960354431274774319053615, −4.40678790073396160840717835838, −4.26080338892158134598736205753, −3.82341485813312170194172909978, −3.82234657133887009705806626236, −3.68454307063217068985201876558, −3.59406520734639204378027772181, −3.42718238953240987128345558460, −3.31707027969180919959931698209, −3.23446433040409682615579724103, −2.94400525068879315617137458881, −2.49237423625452861861361653763, −2.26516666178519712826148479108, −2.15338723956523900867783125274, −1.97660272491721220162874227280, −1.33159854890617141344374262893, 1.33159854890617141344374262893, 1.97660272491721220162874227280, 2.15338723956523900867783125274, 2.26516666178519712826148479108, 2.49237423625452861861361653763, 2.94400525068879315617137458881, 3.23446433040409682615579724103, 3.31707027969180919959931698209, 3.42718238953240987128345558460, 3.59406520734639204378027772181, 3.68454307063217068985201876558, 3.82234657133887009705806626236, 3.82341485813312170194172909978, 4.26080338892158134598736205753, 4.40678790073396160840717835838, 4.62355960354431274774319053615, 5.00543432114936526712372385595, 5.08694410906755579888508315051, 5.15115860058602761065731726403, 5.32768155854376397828334576510, 5.52733087696146860919872411262, 5.66915721391702084461007968095, 5.83216817017281617460988907209, 5.85892989472501405165558560314, 6.15209028999264814683207120899
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.