Dirichlet series
| L(s) = 1 | + 2·2-s + 3-s + 4-s + 2·6-s + 8·7-s + 9-s + 6·11-s + 12-s − 4·13-s + 16·14-s + 24·17-s + 2·18-s − 4·19-s + 8·21-s + 12·22-s − 12·23-s + 5·25-s − 8·26-s + 8·28-s − 15·29-s − 29·31-s − 2·32-s + 6·33-s + 48·34-s + 36-s + 8·37-s − 8·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1/2·4-s + 0.816·6-s + 3.02·7-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 1.10·13-s + 4.27·14-s + 5.82·17-s + 0.471·18-s − 0.917·19-s + 1.74·21-s + 2.55·22-s − 2.50·23-s + 25-s − 1.56·26-s + 1.51·28-s − 2.78·29-s − 5.20·31-s − 0.353·32-s + 1.04·33-s + 8.23·34-s + 1/6·36-s + 1.31·37-s − 1.29·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(23.6766\) |
| Root analytic conductor: | \(1.21869\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.088655210\) |
| \(L(\frac12)\) | \(\approx\) | \(8.088655210\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) | |
| 31 | \( 1 + 29 T + 420 T^{2} + 3931 T^{3} + 25859 T^{4} + 3931 p T^{5} + 420 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 - p T^{2} - 4 p T^{4} + p^{2} T^{6} + 31 p^{2} T^{8} + p^{4} T^{10} - 4 p^{5} T^{12} - p^{7} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 8 T + 37 T^{2} - 136 T^{3} + 44 p T^{4} - 194 T^{5} - 34 p^{2} T^{6} + 9992 T^{7} - 32927 T^{8} + 9992 p T^{9} - 34 p^{4} T^{10} - 194 p^{3} T^{11} + 44 p^{5} T^{12} - 136 p^{5} T^{13} + 37 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 6 T + p T^{2} - 72 T^{3} + 492 T^{4} - 1416 T^{5} + 4522 T^{6} - 1902 p T^{7} + 76961 T^{8} - 1902 p^{2} T^{9} + 4522 p^{2} T^{10} - 1416 p^{3} T^{11} + 492 p^{4} T^{12} - 72 p^{5} T^{13} + p^{7} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 4 T + p T^{2} - 88 T^{3} - 532 T^{4} - 1928 T^{5} - 109 T^{6} + 23966 T^{7} + 124003 T^{8} + 23966 p T^{9} - 109 p^{2} T^{10} - 1928 p^{3} T^{11} - 532 p^{4} T^{12} - 88 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 24 T + 257 T^{2} - 1500 T^{3} + 4260 T^{4} + 1674 T^{5} - 51209 T^{6} + 114960 T^{7} - 48301 T^{8} + 114960 p T^{9} - 51209 p^{2} T^{10} + 1674 p^{3} T^{11} + 4260 p^{4} T^{12} - 1500 p^{5} T^{13} + 257 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 4 T + p T^{2} - 136 T^{3} - 844 T^{4} - 3848 T^{5} - 1219 T^{6} + 67682 T^{7} + 378907 T^{8} + 67682 p T^{9} - 1219 p^{2} T^{10} - 3848 p^{3} T^{11} - 844 p^{4} T^{12} - 136 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 6 T + 13 T^{2} - 60 T^{3} - 659 T^{4} - 60 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 15 T + 32 T^{2} - 825 T^{3} - 6282 T^{4} + 240 T^{5} + 158704 T^{6} + 321750 T^{7} - 1554985 T^{8} + 321750 p T^{9} + 158704 p^{2} T^{10} + 240 p^{3} T^{11} - 6282 p^{4} T^{12} - 825 p^{5} T^{13} + 32 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T - 68 T^{2} + 704 T^{3} + 3338 T^{4} - 35984 T^{5} - 54736 T^{6} + 544472 T^{7} + 1749283 T^{8} + 544472 p T^{9} - 54736 p^{2} T^{10} - 35984 p^{3} T^{11} + 3338 p^{4} T^{12} + 704 p^{5} T^{13} - 68 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 48 T + 1121 T^{2} + 16902 T^{3} + 183756 T^{4} + 1528194 T^{5} + 10236499 T^{6} + 1471056 p T^{7} + 363645887 T^{8} + 1471056 p^{2} T^{9} + 10236499 p^{2} T^{10} + 1528194 p^{3} T^{11} + 183756 p^{4} T^{12} + 16902 p^{5} T^{13} + 1121 p^{6} T^{14} + 48 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 24 T + 243 T^{2} + 622 T^{3} - 13032 T^{4} - 157778 T^{5} - 483959 T^{6} + 5092986 T^{7} + 62275003 T^{8} + 5092986 p T^{9} - 483959 p^{2} T^{10} - 157778 p^{3} T^{11} - 13032 p^{4} T^{12} + 622 p^{5} T^{13} + 243 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 30 T + 386 T^{2} - 3270 T^{3} + 621 p T^{4} - 302070 T^{5} + 2641108 T^{6} - 17866800 T^{7} + 114268505 T^{8} - 17866800 p T^{9} + 2641108 p^{2} T^{10} - 302070 p^{3} T^{11} + 621 p^{5} T^{12} - 3270 p^{5} T^{13} + 386 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 33 T + 413 T^{2} - 870 T^{3} - 40080 T^{4} + 527553 T^{5} - 1835456 T^{6} - 21267300 T^{7} + 290898989 T^{8} - 21267300 p T^{9} - 1835456 p^{2} T^{10} + 527553 p^{3} T^{11} - 40080 p^{4} T^{12} - 870 p^{5} T^{13} + 413 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 3 T + 14 T^{2} + 189 T^{3} - 3243 T^{4} - 11727 T^{5} - 22082 T^{6} - 759081 T^{7} + 7616876 T^{8} - 759081 p T^{9} - 22082 p^{2} T^{10} - 11727 p^{3} T^{11} - 3243 p^{4} T^{12} + 189 p^{5} T^{13} + 14 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 + 14 T + 240 T^{2} + 2026 T^{3} + 20414 T^{4} + 2026 p T^{5} + 240 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 67 | \( 1 - 2 T - 188 T^{2} + 116 T^{3} + 18638 T^{4} + 6154 T^{5} - 1650616 T^{6} - 324802 T^{7} + 126796063 T^{8} - 324802 p T^{9} - 1650616 p^{2} T^{10} + 6154 p^{3} T^{11} + 18638 p^{4} T^{12} + 116 p^{5} T^{13} - 188 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 30 T + 431 T^{2} - 60 p T^{3} + 30840 T^{4} - 147600 T^{5} + 344209 T^{6} + 660 p^{2} T^{7} - 55481941 T^{8} + 660 p^{3} T^{9} + 344209 p^{2} T^{10} - 147600 p^{3} T^{11} + 30840 p^{4} T^{12} - 60 p^{6} T^{13} + 431 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 26 T + 313 T^{2} - 2338 T^{3} + 23948 T^{4} - 349928 T^{5} + 3905291 T^{6} - 29672524 T^{7} + 220000303 T^{8} - 29672524 p T^{9} + 3905291 p^{2} T^{10} - 349928 p^{3} T^{11} + 23948 p^{4} T^{12} - 2338 p^{5} T^{13} + 313 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 189 T^{2} - 710 T^{3} + 17760 T^{4} - 121430 T^{5} + 1498381 T^{6} - 11256060 T^{7} + 121033579 T^{8} - 11256060 p T^{9} + 1498381 p^{2} T^{10} - 121430 p^{3} T^{11} + 17760 p^{4} T^{12} - 710 p^{5} T^{13} + 189 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 - 39 T + 743 T^{2} - 8712 T^{3} + 64248 T^{4} - 157437 T^{5} - 3806684 T^{6} + 77070564 T^{7} - 857329777 T^{8} + 77070564 p T^{9} - 3806684 p^{2} T^{10} - 157437 p^{3} T^{11} + 64248 p^{4} T^{12} - 8712 p^{5} T^{13} + 743 p^{6} T^{14} - 39 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 48 T + 1130 T^{2} - 17730 T^{3} + 199935 T^{4} - 1484904 T^{5} + 3854452 T^{6} + 64956240 T^{7} - 1015209295 T^{8} + 64956240 p T^{9} + 3854452 p^{2} T^{10} - 1484904 p^{3} T^{11} + 199935 p^{4} T^{12} - 17730 p^{5} T^{13} + 1130 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 18 T + 249 T^{2} - 1118 T^{3} - 5772 T^{4} + 231652 T^{5} - 1836929 T^{6} + 6594036 T^{7} + 29614423 T^{8} + 6594036 p T^{9} - 1836929 p^{2} T^{10} + 231652 p^{3} T^{11} - 5772 p^{4} T^{12} - 1118 p^{5} T^{13} + 249 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.63184215096691523900036390862, −5.44985043108868720833883150188, −5.35412437144528816060227935443, −5.20472370610334167731499298520, −5.07043253583315232658703546779, −4.96725668000211701759342568051, −4.94429418567302939907106719833, −4.92384496532247147732811216745, −4.73904347692009456552378970665, −3.92577849688403410800816893035, −3.91646254240753951493073559698, −3.79694358263407670362769158244, −3.74566622152140132278035719941, −3.72203346127693187154516624216, −3.68246110327457266330742654510, −3.46474522662023914138701055587, −3.35255711526274979952868604394, −2.80773708483590754187880986360, −2.39305172493329534181576512735, −2.12214669556417009960019913563, −1.97453338138808900745660137169, −1.75314505348037750588731924866, −1.69326271673494629585947511231, −1.48108711313191562157327269365, −0.938560944752577864538628985285, 0.938560944752577864538628985285, 1.48108711313191562157327269365, 1.69326271673494629585947511231, 1.75314505348037750588731924866, 1.97453338138808900745660137169, 2.12214669556417009960019913563, 2.39305172493329534181576512735, 2.80773708483590754187880986360, 3.35255711526274979952868604394, 3.46474522662023914138701055587, 3.68246110327457266330742654510, 3.72203346127693187154516624216, 3.74566622152140132278035719941, 3.79694358263407670362769158244, 3.91646254240753951493073559698, 3.92577849688403410800816893035, 4.73904347692009456552378970665, 4.92384496532247147732811216745, 4.94429418567302939907106719833, 4.96725668000211701759342568051, 5.07043253583315232658703546779, 5.20472370610334167731499298520, 5.35412437144528816060227935443, 5.44985043108868720833883150188, 5.63184215096691523900036390862
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.