Dirichlet series
| L(s) = 1 | − 2·2-s − 3-s + 4-s + 4·5-s + 2·6-s + 2·7-s + 9-s − 8·10-s − 23·11-s − 12-s − 17·13-s − 4·14-s − 4·15-s + 11·17-s − 2·18-s + 4·20-s − 2·21-s + 46·22-s − 2·23-s + 6·25-s + 34·26-s + 2·28-s + 10·29-s + 8·30-s + 14·31-s + 2·32-s + 23·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.816·6-s + 0.755·7-s + 1/3·9-s − 2.52·10-s − 6.93·11-s − 0.288·12-s − 4.71·13-s − 1.06·14-s − 1.03·15-s + 2.66·17-s − 0.471·18-s + 0.894·20-s − 0.436·21-s + 9.80·22-s − 0.417·23-s + 6/5·25-s + 6.66·26-s + 0.377·28-s + 1.85·29-s + 1.46·30-s + 2.51·31-s + 0.353·32-s + 4.00·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.24869\times 10^{6}\) |
| Root analytic conductor: | \(2.72508\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.0008216552653\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0008216552653\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 3 | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) | |
| 5 | \( ( 1 - T + T^{2} )^{4} \) | |
| 31 | \( 1 - 14 T + 165 T^{2} - 1306 T^{3} + 8279 T^{4} - 1306 p T^{5} + 165 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 7 | \( 1 - 2 T + p T^{2} - 4 T^{3} - 52 T^{4} + 124 T^{5} - 211 T^{6} + 578 T^{7} + 1663 T^{8} + 578 p T^{9} - 211 p^{2} T^{10} + 124 p^{3} T^{11} - 52 p^{4} T^{12} - 4 p^{5} T^{13} + p^{7} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 23 T + 276 T^{2} + 207 p T^{3} + 14471 T^{4} + 75489 T^{5} + 336704 T^{6} + 1319651 T^{7} + 4618872 T^{8} + 1319651 p T^{9} + 336704 p^{2} T^{10} + 75489 p^{3} T^{11} + 14471 p^{4} T^{12} + 207 p^{6} T^{13} + 276 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 17 T + 148 T^{2} + 835 T^{3} + 3560 T^{4} + 1076 p T^{5} + 61214 T^{6} + 21520 p T^{7} + 1116319 T^{8} + 21520 p^{2} T^{9} + 61214 p^{2} T^{10} + 1076 p^{4} T^{11} + 3560 p^{4} T^{12} + 835 p^{5} T^{13} + 148 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 11 T + 77 T^{2} - 250 T^{3} + 320 T^{4} + 2651 T^{5} - 11184 T^{6} + 30400 T^{7} - 2031 T^{8} + 30400 p T^{9} - 11184 p^{2} T^{10} + 2651 p^{3} T^{11} + 320 p^{4} T^{12} - 250 p^{5} T^{13} + 77 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 39 T^{2} - 110 T^{3} + 480 T^{4} - 4400 T^{5} + 7681 T^{6} - 65670 T^{7} + 223399 T^{8} - 65670 p T^{9} + 7681 p^{2} T^{10} - 4400 p^{3} T^{11} + 480 p^{4} T^{12} - 110 p^{5} T^{13} + 39 p^{6} T^{14} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T - 43 T^{2} - 24 T^{3} + 62 p T^{4} + 942 T^{5} - 1475 p T^{6} + 158 T^{7} + 895759 T^{8} + 158 p T^{9} - 1475 p^{3} T^{10} + 942 p^{3} T^{11} + 62 p^{5} T^{12} - 24 p^{5} T^{13} - 43 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 10 T - 28 T^{2} + 750 T^{3} - 2147 T^{4} - 21210 T^{5} + 155504 T^{6} + 224900 T^{7} - 5418335 T^{8} + 224900 p T^{9} + 155504 p^{2} T^{10} - 21210 p^{3} T^{11} - 2147 p^{4} T^{12} + 750 p^{5} T^{13} - 28 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + T - 33 T^{2} - 40 T^{3} + 1310 T^{4} + 3009 T^{5} + 99586 T^{6} - 80 p T^{7} - 3369591 T^{8} - 80 p^{2} T^{9} + 99586 p^{2} T^{10} + 3009 p^{3} T^{11} + 1310 p^{4} T^{12} - 40 p^{5} T^{13} - 33 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 42 T + 901 T^{2} + 318 p T^{3} + 143256 T^{4} + 1280646 T^{5} + 9820379 T^{6} + 67920294 T^{7} + 443297567 T^{8} + 67920294 p T^{9} + 9820379 p^{2} T^{10} + 1280646 p^{3} T^{11} + 143256 p^{4} T^{12} + 318 p^{6} T^{13} + 901 p^{6} T^{14} + 42 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 23 T + 343 T^{2} + 3940 T^{3} + 39200 T^{4} + 348647 T^{5} + 2801414 T^{6} + 20516620 T^{7} + 139083799 T^{8} + 20516620 p T^{9} + 2801414 p^{2} T^{10} + 348647 p^{3} T^{11} + 39200 p^{4} T^{12} + 3940 p^{5} T^{13} + 343 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 11 T + 18 T^{2} + 93 T^{3} + 386 T^{4} + 3864 T^{5} - 18310 T^{6} + 286144 T^{7} - 4368591 T^{8} + 286144 p T^{9} - 18310 p^{2} T^{10} + 3864 p^{3} T^{11} + 386 p^{4} T^{12} + 93 p^{5} T^{13} + 18 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 2 T + 113 T^{2} - 1050 T^{3} + 6220 T^{4} - 106458 T^{5} + 649499 T^{6} - 4812140 T^{7} + 52512019 T^{8} - 4812140 p T^{9} + 649499 p^{2} T^{10} - 106458 p^{3} T^{11} + 6220 p^{4} T^{12} - 1050 p^{5} T^{13} + 113 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 28 T + 449 T^{2} - 5364 T^{3} + 49102 T^{4} - 355728 T^{5} + 1970753 T^{6} - 8504374 T^{7} + 45658651 T^{8} - 8504374 p T^{9} + 1970753 p^{2} T^{10} - 355728 p^{3} T^{11} + 49102 p^{4} T^{12} - 5364 p^{5} T^{13} + 449 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 + 2 T + 48 T^{2} + 454 T^{3} + 3710 T^{4} + 454 p T^{5} + 48 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 67 | \( 1 + 20 T + 142 T^{2} + 1240 T^{3} + 11225 T^{4} - 8560 T^{5} - 632338 T^{6} - 6950180 T^{7} - 75266396 T^{8} - 6950180 p T^{9} - 632338 p^{2} T^{10} - 8560 p^{3} T^{11} + 11225 p^{4} T^{12} + 1240 p^{5} T^{13} + 142 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 42 T + 911 T^{2} - 13068 T^{3} + 132216 T^{4} - 895656 T^{5} + 2652409 T^{6} + 21437496 T^{7} - 338044693 T^{8} + 21437496 p T^{9} + 2652409 p^{2} T^{10} - 895656 p^{3} T^{11} + 132216 p^{4} T^{12} - 13068 p^{5} T^{13} + 911 p^{6} T^{14} - 42 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 4 T + 313 T^{2} - 1052 T^{3} + 51188 T^{4} - 127102 T^{5} + 5549471 T^{6} - 10572716 T^{7} + 455746483 T^{8} - 10572716 p T^{9} + 5549471 p^{2} T^{10} - 127102 p^{3} T^{11} + 51188 p^{4} T^{12} - 1052 p^{5} T^{13} + 313 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 42 T + 819 T^{2} + 11116 T^{3} + 126552 T^{4} + 1187932 T^{5} + 9053398 T^{6} + 65793066 T^{7} + 543610081 T^{8} + 65793066 p T^{9} + 9053398 p^{2} T^{10} + 1187932 p^{3} T^{11} + 126552 p^{4} T^{12} + 11116 p^{5} T^{13} + 819 p^{6} T^{14} + 42 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 6 T + 43 T^{2} - 408 T^{3} + 108 T^{4} + 54372 T^{5} - 211639 T^{6} + 3084546 T^{7} - 79986097 T^{8} + 3084546 p T^{9} - 211639 p^{2} T^{10} + 54372 p^{3} T^{11} + 108 p^{4} T^{12} - 408 p^{5} T^{13} + 43 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 16 T + 114 T^{2} - 1626 T^{3} + 20591 T^{4} - 165528 T^{5} + 2176556 T^{6} - 26354608 T^{7} + 231349137 T^{8} - 26354608 p T^{9} + 2176556 p^{2} T^{10} - 165528 p^{3} T^{11} + 20591 p^{4} T^{12} - 1626 p^{5} T^{13} + 114 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 - 22 T + 87 T^{2} + 1390 T^{3} - 20839 T^{4} + 1390 p T^{5} + 87 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
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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
−4.42838594351117995738589841399, −4.26006654697124951799974830207, −4.24319446413991671362278572151, −3.91213771195508048067200429280, −3.79450462112910842544379416651, −3.58225641215675806587297469259, −3.31434202199306548894198281187, −3.14973708180989475823857475922, −3.09819808930790112772603759538, −2.96448613894639466814874919646, −2.75745010284648526187861814438, −2.72666174832766637764434283343, −2.61329671477653158470206579974, −2.42927895160789370177565034204, −2.36901326351467650347504069951, −2.24964910288865027582020878688, −2.22335613455923743780295569349, −1.76105633305243695212699564963, −1.66500658819474008650629694797, −1.49535069604546569610209985093, −1.28158618085490548207833998856, −0.961970457176416656232490523281, −0.58969512752545236508461933107, −0.07038077519518455138621670679, −0.04957703186415566341201908032, 0.04957703186415566341201908032, 0.07038077519518455138621670679, 0.58969512752545236508461933107, 0.961970457176416656232490523281, 1.28158618085490548207833998856, 1.49535069604546569610209985093, 1.66500658819474008650629694797, 1.76105633305243695212699564963, 2.22335613455923743780295569349, 2.24964910288865027582020878688, 2.36901326351467650347504069951, 2.42927895160789370177565034204, 2.61329671477653158470206579974, 2.72666174832766637764434283343, 2.75745010284648526187861814438, 2.96448613894639466814874919646, 3.09819808930790112772603759538, 3.14973708180989475823857475922, 3.31434202199306548894198281187, 3.58225641215675806587297469259, 3.79450462112910842544379416651, 3.91213771195508048067200429280, 4.24319446413991671362278572151, 4.26006654697124951799974830207, 4.42838594351117995738589841399
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.