Dirichlet series
| L(s) = 1 | − 6·3-s + 4·5-s − 2·7-s + 14·9-s − 6·11-s + 24·13-s − 24·15-s + 12·17-s + 6·19-s + 12·21-s − 6·23-s + 8·25-s − 12·27-s − 12·31-s + 36·33-s − 8·35-s − 144·39-s − 4·41-s − 22·43-s + 56·45-s + 8·47-s + 2·49-s − 72·51-s − 48·53-s − 24·55-s − 36·57-s + 22·59-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 1.78·5-s − 0.755·7-s + 14/3·9-s − 1.80·11-s + 6.65·13-s − 6.19·15-s + 2.91·17-s + 1.37·19-s + 2.61·21-s − 1.25·23-s + 8/5·25-s − 2.30·27-s − 2.15·31-s + 6.26·33-s − 1.35·35-s − 23.0·39-s − 0.624·41-s − 3.35·43-s + 8.34·45-s + 1.16·47-s + 2/7·49-s − 10.0·51-s − 6.59·53-s − 3.23·55-s − 4.76·57-s + 2.86·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{40} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(14823.9\) |
| Root analytic conductor: | \(1.82257\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{40} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.280031737\) |
| \(L(\frac12)\) | \(\approx\) | \(1.280031737\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 - 6 T + p T^{2} )^{4} \) | |
| good | 3 | \( 1 + 2 p T + 22 T^{2} + 20 p T^{3} + 43 p T^{4} + 28 p^{2} T^{5} + 466 T^{6} + 94 p^{2} T^{7} + 1516 T^{8} + 94 p^{3} T^{9} + 466 p^{2} T^{10} + 28 p^{5} T^{11} + 43 p^{5} T^{12} + 20 p^{6} T^{13} + 22 p^{6} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 4 T^{4} + 76 T^{5} - 8 p^{2} T^{6} + 84 p T^{7} - 794 T^{8} + 84 p^{2} T^{9} - 8 p^{4} T^{10} + 76 p^{3} T^{11} - 4 p^{4} T^{12} - 12 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 2 T + 2 T^{2} + 4 p T^{3} + 101 T^{4} + 4 p^{2} T^{5} + 582 T^{6} + 2202 T^{7} + 5716 T^{8} + 2202 p T^{9} + 582 p^{2} T^{10} + 4 p^{5} T^{11} + 101 p^{4} T^{12} + 4 p^{6} T^{13} + 2 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 6 T + 30 T^{2} + 84 T^{3} + 145 T^{4} - 348 T^{5} - 3030 T^{6} - 17202 T^{7} - 56916 T^{8} - 17202 p T^{9} - 3030 p^{2} T^{10} - 348 p^{3} T^{11} + 145 p^{4} T^{12} + 84 p^{5} T^{13} + 30 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 12 T + 114 T^{2} - 792 T^{3} + 4693 T^{4} - 23520 T^{5} + 110490 T^{6} - 471996 T^{7} + 1997436 T^{8} - 471996 p T^{9} + 110490 p^{2} T^{10} - 23520 p^{3} T^{11} + 4693 p^{4} T^{12} - 792 p^{5} T^{13} + 114 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 6 T + 30 T^{2} - 252 T^{3} + 1489 T^{4} - 7620 T^{5} + 40746 T^{6} - 188694 T^{7} + 823116 T^{8} - 188694 p T^{9} + 40746 p^{2} T^{10} - 7620 p^{3} T^{11} + 1489 p^{4} T^{12} - 252 p^{5} T^{13} + 30 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 6 T - 2 T^{2} - 156 T^{3} - 1151 T^{4} - 5604 T^{5} - 4262 T^{6} + 135054 T^{7} + 1066684 T^{8} + 135054 p T^{9} - 4262 p^{2} T^{10} - 5604 p^{3} T^{11} - 1151 p^{4} T^{12} - 156 p^{5} T^{13} - 2 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 38 T^{2} + 144 T^{3} - 59 T^{4} - 4824 T^{5} + 11986 T^{6} + 50472 T^{7} + 186988 T^{8} + 50472 p T^{9} + 11986 p^{2} T^{10} - 4824 p^{3} T^{11} - 59 p^{4} T^{12} + 144 p^{5} T^{13} - 38 p^{6} T^{14} + p^{8} T^{16} \) | |
| 31 | \( 1 + 12 T + 72 T^{2} + 540 T^{3} + 5500 T^{4} + 36684 T^{5} + 190008 T^{6} + 1281084 T^{7} + 8521734 T^{8} + 1281084 p T^{9} + 190008 p^{2} T^{10} + 36684 p^{3} T^{11} + 5500 p^{4} T^{12} + 540 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 90 T^{2} - 96 T^{3} + 2929 T^{4} - 12480 T^{5} + 25218 T^{6} - 933600 T^{7} - 881004 T^{8} - 933600 p T^{9} + 25218 p^{2} T^{10} - 12480 p^{3} T^{11} + 2929 p^{4} T^{12} - 96 p^{5} T^{13} + 90 p^{6} T^{14} + p^{8} T^{16} \) | |
| 41 | \( 1 + 4 T - 10 T^{2} - 240 T^{3} - 1339 T^{4} - 2584 T^{5} + 24694 T^{6} + 292068 T^{7} - 192452 T^{8} + 292068 p T^{9} + 24694 p^{2} T^{10} - 2584 p^{3} T^{11} - 1339 p^{4} T^{12} - 240 p^{5} T^{13} - 10 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 22 T + 150 T^{2} + 788 T^{3} + 14033 T^{4} + 120588 T^{5} + 419410 T^{6} + 4311526 T^{7} + 47947500 T^{8} + 4311526 p T^{9} + 419410 p^{2} T^{10} + 120588 p^{3} T^{11} + 14033 p^{4} T^{12} + 788 p^{5} T^{13} + 150 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( ( 1 - 4 T + 8 T^{2} - 100 T^{3} + 766 T^{4} - 100 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 53 | \( ( 1 + 24 T + 368 T^{2} + 72 p T^{3} + 31806 T^{4} + 72 p^{2} T^{5} + 368 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 - 22 T + 410 T^{2} - 4596 T^{3} + 46661 T^{4} - 322124 T^{5} + 2140414 T^{6} - 9312390 T^{7} + 69548260 T^{8} - 9312390 p T^{9} + 2140414 p^{2} T^{10} - 322124 p^{3} T^{11} + 46661 p^{4} T^{12} - 4596 p^{5} T^{13} + 410 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 4 T - 138 T^{2} + 472 T^{3} + 8177 T^{4} - 10056 T^{5} - 748634 T^{6} - 248860 T^{7} + 65687988 T^{8} - 248860 p T^{9} - 748634 p^{2} T^{10} - 10056 p^{3} T^{11} + 8177 p^{4} T^{12} + 472 p^{5} T^{13} - 138 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 2 T - 22 T^{2} + 596 T^{3} - 955 T^{4} - 1540 T^{5} + 155118 T^{6} - 1149330 T^{7} + 1782532 T^{8} - 1149330 p T^{9} + 155118 p^{2} T^{10} - 1540 p^{3} T^{11} - 955 p^{4} T^{12} + 596 p^{5} T^{13} - 22 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 14 T + 242 T^{2} + 1956 T^{3} + 15989 T^{4} - 3428 T^{5} - 834218 T^{6} - 18859242 T^{7} - 160743884 T^{8} - 18859242 p T^{9} - 834218 p^{2} T^{10} - 3428 p^{3} T^{11} + 15989 p^{4} T^{12} + 1956 p^{5} T^{13} + 242 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 12 T + 72 T^{2} - 396 T^{3} - 7076 T^{4} - 25668 T^{5} + 279864 T^{6} + 1767300 T^{7} + 23147910 T^{8} + 1767300 p T^{9} + 279864 p^{2} T^{10} - 25668 p^{3} T^{11} - 7076 p^{4} T^{12} - 396 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 224 T^{2} + 30940 T^{4} - 3015968 T^{6} + 249183430 T^{8} - 3015968 p^{2} T^{10} + 30940 p^{4} T^{12} - 224 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 - 44 T + 968 T^{2} - 15660 T^{3} + 206492 T^{4} - 2233516 T^{5} + 21008248 T^{6} - 186069324 T^{7} + 1657961446 T^{8} - 186069324 p T^{9} + 21008248 p^{2} T^{10} - 2233516 p^{3} T^{11} + 206492 p^{4} T^{12} - 15660 p^{5} T^{13} + 968 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 32 T + 458 T^{2} + 1848 T^{3} - 39487 T^{4} - 715376 T^{5} - 3565454 T^{6} + 33394392 T^{7} + 654295780 T^{8} + 33394392 p T^{9} - 3565454 p^{2} T^{10} - 715376 p^{3} T^{11} - 39487 p^{4} T^{12} + 1848 p^{5} T^{13} + 458 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 68 T + 1862 T^{2} - 22264 T^{3} - 16651 T^{4} + 3698264 T^{5} - 33762042 T^{6} - 152904156 T^{7} + 4581782908 T^{8} - 152904156 p T^{9} - 33762042 p^{2} T^{10} + 3698264 p^{3} T^{11} - 16651 p^{4} T^{12} - 22264 p^{5} T^{13} + 1862 p^{6} T^{14} - 68 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.19070343920528765683141300868, −4.95985794968096808861676130563, −4.84626610922084159777285236790, −4.78108041171022257978986034173, −4.59100384857136812410330363218, −4.25455343496039020146304197730, −4.02205998338317954254524574932, −3.94089780944467549905547808420, −3.58005204612683143344344336354, −3.57258443076229461536976002561, −3.38910595712710684130421419801, −3.33025575084907479273667016818, −3.28973872724973852447483416302, −3.14632230242505281418508542330, −3.12220261457826960377958997611, −2.65912322804041651295669807508, −2.13800153624399432219774384029, −1.97036190305330429753729176399, −1.95329255781088577836564254144, −1.71773422448241610247812015253, −1.32966464616894399132730820979, −1.26890940056556601935506965929, −1.07862855370893804093966555214, −0.66876770173187298434773279440, −0.39861124710530090784570850206, 0.39861124710530090784570850206, 0.66876770173187298434773279440, 1.07862855370893804093966555214, 1.26890940056556601935506965929, 1.32966464616894399132730820979, 1.71773422448241610247812015253, 1.95329255781088577836564254144, 1.97036190305330429753729176399, 2.13800153624399432219774384029, 2.65912322804041651295669807508, 3.12220261457826960377958997611, 3.14632230242505281418508542330, 3.28973872724973852447483416302, 3.33025575084907479273667016818, 3.38910595712710684130421419801, 3.57258443076229461536976002561, 3.58005204612683143344344336354, 3.94089780944467549905547808420, 4.02205998338317954254524574932, 4.25455343496039020146304197730, 4.59100384857136812410330363218, 4.78108041171022257978986034173, 4.84626610922084159777285236790, 4.95985794968096808861676130563, 5.19070343920528765683141300868
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.