Dirichlet series
| L(s) = 1 | − 5·3-s − 7·5-s + 8·7-s + 4·9-s + 8·11-s + 8·13-s + 35·15-s + 3·17-s − 19·19-s − 40·21-s + 23-s + 12·25-s + 23·27-s − 21·29-s + 2·31-s − 40·33-s − 56·35-s − 12·37-s − 40·39-s − 6·41-s − 19·43-s − 28·45-s − 7·47-s + 36·49-s − 15·51-s − 26·53-s − 56·55-s + ⋯ |
| L(s) = 1 | − 2.88·3-s − 3.13·5-s + 3.02·7-s + 4/3·9-s + 2.41·11-s + 2.21·13-s + 9.03·15-s + 0.727·17-s − 4.35·19-s − 8.72·21-s + 0.208·23-s + 12/5·25-s + 4.42·27-s − 3.89·29-s + 0.359·31-s − 6.96·33-s − 9.46·35-s − 1.97·37-s − 6.40·39-s − 0.937·41-s − 2.89·43-s − 4.17·45-s − 1.02·47-s + 36/7·49-s − 2.10·51-s − 3.57·53-s − 7.55·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 7^{8} \cdot 11^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.79517\times 10^{14}\) |
| Root analytic conductor: | \(7.99651\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{24} \cdot 7^{8} \cdot 11^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - T )^{8} \) | |
| 11 | \( ( 1 - T )^{8} \) | |
| 13 | \( ( 1 - T )^{8} \) | |
| good | 3 | \( 1 + 5 T + 7 p T^{2} + 62 T^{3} + 56 p T^{4} + 43 p^{2} T^{5} + 844 T^{6} + 1634 T^{7} + 997 p T^{8} + 1634 p T^{9} + 844 p^{2} T^{10} + 43 p^{5} T^{11} + 56 p^{5} T^{12} + 62 p^{5} T^{13} + 7 p^{7} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 7 T + 37 T^{2} + 138 T^{3} + 94 p T^{4} + 1387 T^{5} + 782 p T^{6} + 9826 T^{7} + 23297 T^{8} + 9826 p T^{9} + 782 p^{3} T^{10} + 1387 p^{3} T^{11} + 94 p^{5} T^{12} + 138 p^{5} T^{13} + 37 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 3 T + 93 T^{2} - 252 T^{3} + 4302 T^{4} - 10279 T^{5} + 126478 T^{6} - 261210 T^{7} + 2561201 T^{8} - 261210 p T^{9} + 126478 p^{2} T^{10} - 10279 p^{3} T^{11} + 4302 p^{4} T^{12} - 252 p^{5} T^{13} + 93 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + p T + 227 T^{2} + 1954 T^{3} + 14128 T^{4} + 87909 T^{5} + 494602 T^{6} + 2495046 T^{7} + 11479137 T^{8} + 2495046 p T^{9} + 494602 p^{2} T^{10} + 87909 p^{3} T^{11} + 14128 p^{4} T^{12} + 1954 p^{5} T^{13} + 227 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - T + 88 T^{2} - 43 T^{3} + 3913 T^{4} - 1631 T^{5} + 125444 T^{6} - 59829 T^{7} + 3200004 T^{8} - 59829 p T^{9} + 125444 p^{2} T^{10} - 1631 p^{3} T^{11} + 3913 p^{4} T^{12} - 43 p^{5} T^{13} + 88 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 21 T + 274 T^{2} + 2507 T^{3} + 18563 T^{4} + 4113 p T^{5} + 24662 p T^{6} + 4125387 T^{7} + 22760696 T^{8} + 4125387 p T^{9} + 24662 p^{3} T^{10} + 4113 p^{4} T^{11} + 18563 p^{4} T^{12} + 2507 p^{5} T^{13} + 274 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 2 T + 92 T^{2} - 92 T^{3} + 4979 T^{4} - 2492 T^{5} + 206564 T^{6} - 93938 T^{7} + 7282920 T^{8} - 93938 p T^{9} + 206564 p^{2} T^{10} - 2492 p^{3} T^{11} + 4979 p^{4} T^{12} - 92 p^{5} T^{13} + 92 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 12 T + 204 T^{2} + 1347 T^{3} + 13057 T^{4} + 1171 p T^{5} + 342344 T^{6} - 172162 T^{7} + 6121444 T^{8} - 172162 p T^{9} + 342344 p^{2} T^{10} + 1171 p^{4} T^{11} + 13057 p^{4} T^{12} + 1347 p^{5} T^{13} + 204 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 6 T + 160 T^{2} + 933 T^{3} + 14271 T^{4} + 1849 p T^{5} + 908892 T^{6} + 4330220 T^{7} + 43151784 T^{8} + 4330220 p T^{9} + 908892 p^{2} T^{10} + 1849 p^{4} T^{11} + 14271 p^{4} T^{12} + 933 p^{5} T^{13} + 160 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 19 T + 6 p T^{2} + 2289 T^{3} + 18927 T^{4} + 137407 T^{5} + 1154882 T^{6} + 8749350 T^{7} + 64231879 T^{8} + 8749350 p T^{9} + 1154882 p^{2} T^{10} + 137407 p^{3} T^{11} + 18927 p^{4} T^{12} + 2289 p^{5} T^{13} + 6 p^{7} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 7 T + 262 T^{2} + 1287 T^{3} + 31427 T^{4} + 121039 T^{5} + 2455918 T^{6} + 170273 p T^{7} + 136663328 T^{8} + 170273 p^{2} T^{9} + 2455918 p^{2} T^{10} + 121039 p^{3} T^{11} + 31427 p^{4} T^{12} + 1287 p^{5} T^{13} + 262 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 26 T + 466 T^{2} + 6604 T^{3} + 78009 T^{4} + 791550 T^{5} + 7255826 T^{6} + 59916664 T^{7} + 453887083 T^{8} + 59916664 p T^{9} + 7255826 p^{2} T^{10} + 791550 p^{3} T^{11} + 78009 p^{4} T^{12} + 6604 p^{5} T^{13} + 466 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 17 T + 437 T^{2} + 5468 T^{3} + 81413 T^{4} + 802050 T^{5} + 8853287 T^{6} + 71237211 T^{7} + 633150684 T^{8} + 71237211 p T^{9} + 8853287 p^{2} T^{10} + 802050 p^{3} T^{11} + 81413 p^{4} T^{12} + 5468 p^{5} T^{13} + 437 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 117 T^{2} + 253 T^{3} + 13510 T^{4} + 29064 T^{5} + 1010996 T^{6} + 2355402 T^{7} + 73020483 T^{8} + 2355402 p T^{9} + 1010996 p^{2} T^{10} + 29064 p^{3} T^{11} + 13510 p^{4} T^{12} + 253 p^{5} T^{13} + 117 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 + 24 T + 571 T^{2} + 9254 T^{3} + 134590 T^{4} + 1627284 T^{5} + 17856782 T^{6} + 169814174 T^{7} + 1487571735 T^{8} + 169814174 p T^{9} + 17856782 p^{2} T^{10} + 1627284 p^{3} T^{11} + 134590 p^{4} T^{12} + 9254 p^{5} T^{13} + 571 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 2 T + 171 T^{2} - 44 T^{3} + 6659 T^{4} + 7580 T^{5} - 470127 T^{6} - 2077898 T^{7} - 62333328 T^{8} - 2077898 p T^{9} - 470127 p^{2} T^{10} + 7580 p^{3} T^{11} + 6659 p^{4} T^{12} - 44 p^{5} T^{13} + 171 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 2 T + 206 T^{2} + 31 T^{3} + 24051 T^{4} + 39147 T^{5} + 29026 p T^{6} + 5259840 T^{7} + 163712008 T^{8} + 5259840 p T^{9} + 29026 p^{3} T^{10} + 39147 p^{3} T^{11} + 24051 p^{4} T^{12} + 31 p^{5} T^{13} + 206 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 7 T + 510 T^{2} - 2915 T^{3} + 119649 T^{4} - 564031 T^{5} + 17024124 T^{6} - 66706738 T^{7} + 1621590849 T^{8} - 66706738 p T^{9} + 17024124 p^{2} T^{10} - 564031 p^{3} T^{11} + 119649 p^{4} T^{12} - 2915 p^{5} T^{13} + 510 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 6 T + 387 T^{2} + 1621 T^{3} + 68918 T^{4} + 161888 T^{5} + 7853966 T^{6} + 8489740 T^{7} + 701863969 T^{8} + 8489740 p T^{9} + 7853966 p^{2} T^{10} + 161888 p^{3} T^{11} + 68918 p^{4} T^{12} + 1621 p^{5} T^{13} + 387 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 13 T + 367 T^{2} - 5350 T^{3} + 85200 T^{4} - 1030769 T^{5} + 13079114 T^{6} - 1493594 p T^{7} + 1373226193 T^{8} - 1493594 p^{2} T^{9} + 13079114 p^{2} T^{10} - 1030769 p^{3} T^{11} + 85200 p^{4} T^{12} - 5350 p^{5} T^{13} + 367 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 18 T + 752 T^{2} - 10531 T^{3} + 246943 T^{4} - 2791331 T^{5} + 46835252 T^{6} - 431230216 T^{7} + 5633846776 T^{8} - 431230216 p T^{9} + 46835252 p^{2} T^{10} - 2791331 p^{3} T^{11} + 246943 p^{4} T^{12} - 10531 p^{5} T^{13} + 752 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 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
−3.68744651225800557902920628007, −3.45879310924429680309827515518, −3.32916375528002342751741831019, −3.27365607639373030708430520203, −3.19301073882954530283910884442, −3.19130957463695629846007457952, −3.16811059016404385167860921953, −3.13237859801300528386401488690, −2.92660099989592088708577260389, −2.41871953741987757502024976059, −2.28969235467651695924398635058, −2.26569973128827656720548098553, −2.26512615425100505508977732884, −2.15638787255446216673960545133, −1.95186565136203796041376299459, −1.89774200074983809096342833236, −1.85662627675763665583787457556, −1.56354921327435210741782192108, −1.48197457008487123710802634863, −1.39401451660083662522470921493, −1.35103875534390519695266869762, −1.12396990776620900929751655106, −1.04508113307637760583549625555, −0.950032455377121021857336612451, −0.920759648226703934855601715602, 0, 0, 0, 0, 0, 0, 0, 0, 0.920759648226703934855601715602, 0.950032455377121021857336612451, 1.04508113307637760583549625555, 1.12396990776620900929751655106, 1.35103875534390519695266869762, 1.39401451660083662522470921493, 1.48197457008487123710802634863, 1.56354921327435210741782192108, 1.85662627675763665583787457556, 1.89774200074983809096342833236, 1.95186565136203796041376299459, 2.15638787255446216673960545133, 2.26512615425100505508977732884, 2.26569973128827656720548098553, 2.28969235467651695924398635058, 2.41871953741987757502024976059, 2.92660099989592088708577260389, 3.13237859801300528386401488690, 3.16811059016404385167860921953, 3.19130957463695629846007457952, 3.19301073882954530283910884442, 3.27365607639373030708430520203, 3.32916375528002342751741831019, 3.45879310924429680309827515518, 3.68744651225800557902920628007
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.