Dirichlet series
| L(s) = 1 | − 4·2-s + 6·4-s + 3·5-s + 5·7-s − 2·9-s − 12·10-s − 3·11-s + 4·13-s − 20·14-s − 15·16-s − 15·17-s + 8·18-s + 9·19-s + 18·20-s + 12·22-s − 6·23-s + 7·25-s − 16·26-s + 30·28-s − 3·29-s + 24·32-s + 60·34-s + 15·35-s − 12·36-s + 20·37-s − 36·38-s + 21·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s + 1.34·5-s + 1.88·7-s − 2/3·9-s − 3.79·10-s − 0.904·11-s + 1.10·13-s − 5.34·14-s − 3.75·16-s − 3.63·17-s + 1.88·18-s + 2.06·19-s + 4.02·20-s + 2.55·22-s − 1.25·23-s + 7/5·25-s − 3.13·26-s + 5.66·28-s − 0.557·29-s + 4.24·32-s + 10.2·34-s + 2.53·35-s − 2·36-s + 3.28·37-s − 5.83·38-s + 3.27·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.34822\) |
| Root analytic conductor: | \(1.01884\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3662945657\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3662945657\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 5 | \( 1 - 3 T + 2 T^{2} + 3 p T^{3} - 54 T^{4} + 3 p^{2} T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| 13 | \( 1 - 4 T + 28 T^{2} - 118 T^{3} + 523 T^{4} - 118 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 + 2 T^{2} - 2 p T^{4} - 2 p T^{5} - 4 T^{6} + 20 p T^{7} + 19 T^{8} + 20 p^{2} T^{9} - 4 p^{2} T^{10} - 2 p^{4} T^{11} - 2 p^{5} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 5 T + 6 T^{2} + 41 T^{3} - 199 T^{4} + 432 T^{5} - 74 T^{6} - 3242 T^{7} + 13092 T^{8} - 3242 p T^{9} - 74 p^{2} T^{10} + 432 p^{3} T^{11} - 199 p^{4} T^{12} + 41 p^{5} T^{13} + 6 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 3 T + 28 T^{2} + 75 T^{3} + 397 T^{4} + 1512 T^{5} + 5050 T^{6} + 23220 T^{7} + 57412 T^{8} + 23220 p T^{9} + 5050 p^{2} T^{10} + 1512 p^{3} T^{11} + 397 p^{4} T^{12} + 75 p^{5} T^{13} + 28 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 15 T + 151 T^{2} + 1140 T^{3} + 7387 T^{4} + 42657 T^{5} + 221164 T^{6} + 1045737 T^{7} + 4488412 T^{8} + 1045737 p T^{9} + 221164 p^{2} T^{10} + 42657 p^{3} T^{11} + 7387 p^{4} T^{12} + 1140 p^{5} T^{13} + 151 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 9 T + 94 T^{2} - 603 T^{3} + 205 p T^{4} - 18144 T^{5} + 93484 T^{6} - 373770 T^{7} + 1769680 T^{8} - 373770 p T^{9} + 93484 p^{2} T^{10} - 18144 p^{3} T^{11} + 205 p^{5} T^{12} - 603 p^{5} T^{13} + 94 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 6 T + 88 T^{2} + 456 T^{3} + 4126 T^{4} + 21306 T^{5} + 141472 T^{6} + 672282 T^{7} + 3633727 T^{8} + 672282 p T^{9} + 141472 p^{2} T^{10} + 21306 p^{3} T^{11} + 4126 p^{4} T^{12} + 456 p^{5} T^{13} + 88 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 3 T - 89 T^{2} - 108 T^{3} + 5023 T^{4} + 1659 T^{5} - 202094 T^{6} - 22485 T^{7} + 6307612 T^{8} - 22485 p T^{9} - 202094 p^{2} T^{10} + 1659 p^{3} T^{11} + 5023 p^{4} T^{12} - 108 p^{5} T^{13} - 89 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 188 T^{2} + 16924 T^{4} - 941624 T^{6} + 35207710 T^{8} - 941624 p^{2} T^{10} + 16924 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} \) | |
| 37 | \( 1 - 20 T + 132 T^{2} - 484 T^{3} + 5861 T^{4} - 55470 T^{5} + 263104 T^{6} - 1941398 T^{7} + 16342164 T^{8} - 1941398 p T^{9} + 263104 p^{2} T^{10} - 55470 p^{3} T^{11} + 5861 p^{4} T^{12} - 484 p^{5} T^{13} + 132 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 21 T + 283 T^{2} - 2856 T^{3} + 23455 T^{4} - 158745 T^{5} + 910306 T^{6} - 5010921 T^{7} + 29380960 T^{8} - 5010921 p T^{9} + 910306 p^{2} T^{10} - 158745 p^{3} T^{11} + 23455 p^{4} T^{12} - 2856 p^{5} T^{13} + 283 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 18 T + 244 T^{2} - 2448 T^{3} + 19486 T^{4} - 115506 T^{5} + 569656 T^{6} - 2068326 T^{7} + 9076111 T^{8} - 2068326 p T^{9} + 569656 p^{2} T^{10} - 115506 p^{3} T^{11} + 19486 p^{4} T^{12} - 2448 p^{5} T^{13} + 244 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( ( 1 - 3 T + 71 T^{2} - 156 T^{3} + 2160 T^{4} - 156 p T^{5} + 71 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 53 | \( 1 - 68 T^{2} + 7318 T^{4} - 456764 T^{6} + 29553859 T^{8} - 456764 p^{2} T^{10} + 7318 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \) | |
| 59 | \( 1 - 30 T + 580 T^{2} - 8400 T^{3} + 101470 T^{4} - 1072734 T^{5} + 10172152 T^{6} - 88229610 T^{7} + 703838287 T^{8} - 88229610 p T^{9} + 10172152 p^{2} T^{10} - 1072734 p^{3} T^{11} + 101470 p^{4} T^{12} - 8400 p^{5} T^{13} + 580 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 5 T + 27 T^{2} + 208 T^{3} + 2873 T^{4} + 16809 T^{5} + 25594 T^{6} + 1324415 T^{7} + 2164560 T^{8} + 1324415 p T^{9} + 25594 p^{2} T^{10} + 16809 p^{3} T^{11} + 2873 p^{4} T^{12} + 208 p^{5} T^{13} + 27 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 71 | \( 1 + 124 T^{2} + 3754 T^{4} - 20352 T^{5} + 240112 T^{6} - 4462848 T^{7} + 35638579 T^{8} - 4462848 p T^{9} + 240112 p^{2} T^{10} - 20352 p^{3} T^{11} + 3754 p^{4} T^{12} + 124 p^{6} T^{14} + p^{8} T^{16} \) | |
| 73 | \( ( 1 - 13 T + 178 T^{2} - 1549 T^{3} + 13924 T^{4} - 1549 p T^{5} + 178 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 79 | \( ( 1 - 2 T + 100 T^{2} + 28 T^{3} + 4702 T^{4} + 28 p T^{5} + 100 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 83 | \( ( 1 + 24 T + 410 T^{2} + 4374 T^{3} + 44538 T^{4} + 4374 p T^{5} + 410 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 39 T + 904 T^{2} + 15483 T^{3} + 210337 T^{4} + 2365440 T^{5} + 23189230 T^{6} + 211685526 T^{7} + 1943350048 T^{8} + 211685526 p T^{9} + 23189230 p^{2} T^{10} + 2365440 p^{3} T^{11} + 210337 p^{4} T^{12} + 15483 p^{5} T^{13} + 904 p^{6} T^{14} + 39 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 4 T - 312 T^{2} - 244 T^{3} + 60230 T^{4} - 23286 T^{5} - 8035652 T^{6} + 909880 T^{7} + 827060415 T^{8} + 909880 p T^{9} - 8035652 p^{2} T^{10} - 23286 p^{3} T^{11} + 60230 p^{4} T^{12} - 244 p^{5} T^{13} - 312 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−6.17912854764054639049387850417, −5.98731337779068116866648918219, −5.78600773030713859183683066074, −5.74870823462508683364763830396, −5.67095026610952465052994156741, −5.57857892518856183264273593148, −5.18793584489183445169454563101, −5.13148422440015051836859926220, −4.95751463253755536683903168223, −4.51487911498066216539448404581, −4.41599723686787318792423401473, −4.28942061007337349723234755658, −4.11749428142288170357503696970, −4.06316575708088113164335987039, −4.01372597711788230737255708628, −3.50446013331580491312432713442, −2.83241364847034310728273070018, −2.76650445435974183825731678520, −2.50570930030457100859793225972, −2.46445432879887936497326973791, −2.27467957173551348400212843428, −1.87954448925902756475773669162, −1.49140672827941370793517788811, −0.999356322060759162187766144789, −0.924146857455589002477308090002, 0.924146857455589002477308090002, 0.999356322060759162187766144789, 1.49140672827941370793517788811, 1.87954448925902756475773669162, 2.27467957173551348400212843428, 2.46445432879887936497326973791, 2.50570930030457100859793225972, 2.76650445435974183825731678520, 2.83241364847034310728273070018, 3.50446013331580491312432713442, 4.01372597711788230737255708628, 4.06316575708088113164335987039, 4.11749428142288170357503696970, 4.28942061007337349723234755658, 4.41599723686787318792423401473, 4.51487911498066216539448404581, 4.95751463253755536683903168223, 5.13148422440015051836859926220, 5.18793584489183445169454563101, 5.57857892518856183264273593148, 5.67095026610952465052994156741, 5.74870823462508683364763830396, 5.78600773030713859183683066074, 5.98731337779068116866648918219, 6.17912854764054639049387850417
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.