Dirichlet series
| L(s) = 1 | + 4·2-s + 3-s + 7·4-s − 2·5-s + 4·6-s + 3·7-s + 8·8-s + 9-s − 8·10-s + 7·12-s − 4·13-s + 12·14-s − 2·15-s + 4·16-s − 17-s + 4·18-s + 19-s − 14·20-s + 3·21-s − 18·23-s + 8·24-s + 25-s − 16·26-s + 3·27-s + 21·28-s − 19·29-s − 8·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.577·3-s + 7/2·4-s − 0.894·5-s + 1.63·6-s + 1.13·7-s + 2.82·8-s + 1/3·9-s − 2.52·10-s + 2.02·12-s − 1.10·13-s + 3.20·14-s − 0.516·15-s + 16-s − 0.242·17-s + 0.942·18-s + 0.229·19-s − 3.13·20-s + 0.654·21-s − 3.75·23-s + 1.63·24-s + 1/5·25-s − 3.13·26-s + 0.577·27-s + 3.96·28-s − 3.52·29-s − 1.46·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(296660.\) |
| Root analytic conductor: | \(2.19794\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.11091238\) |
| \(L(\frac12)\) | \(\approx\) | \(10.11091238\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - p^{2} T + 9 T^{2} - p^{4} T^{3} + 29 T^{4} - 7 p^{3} T^{5} + 93 T^{6} - p^{7} T^{7} + 173 T^{8} - p^{8} T^{9} + 93 p^{2} T^{10} - 7 p^{6} T^{11} + 29 p^{4} T^{12} - p^{9} T^{13} + 9 p^{6} T^{14} - p^{9} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - T - 2 T^{3} + 2 p T^{4} - 29 T^{5} + 34 T^{6} - 20 T^{7} + 79 T^{8} - 20 p T^{9} + 34 p^{2} T^{10} - 29 p^{3} T^{11} + 2 p^{5} T^{12} - 2 p^{5} T^{13} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 3 T + 5 T^{2} + 11 T^{3} - 29 T^{4} - 62 T^{5} + 96 T^{6} + 690 T^{7} - 5491 T^{8} + 690 p T^{9} + 96 p^{2} T^{10} - 62 p^{3} T^{11} - 29 p^{4} T^{12} + 11 p^{5} T^{13} + 5 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 4 T + 7 T^{2} + 59 T^{3} + 437 T^{4} + 1241 T^{5} + 2351 T^{6} + 18028 T^{7} + 93877 T^{8} + 18028 p T^{9} + 2351 p^{2} T^{10} + 1241 p^{3} T^{11} + 437 p^{4} T^{12} + 59 p^{5} T^{13} + 7 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + T - 36 T^{2} + 29 T^{3} + 574 T^{4} - 716 T^{5} + 2868 T^{6} + 666 p T^{7} - 138837 T^{8} + 666 p^{2} T^{9} + 2868 p^{2} T^{10} - 716 p^{3} T^{11} + 574 p^{4} T^{12} + 29 p^{5} T^{13} - 36 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - T - 7 T^{2} + 96 T^{3} + 293 T^{4} + 994 T^{5} - 1110 T^{6} + 17535 T^{7} + 246561 T^{8} + 17535 p T^{9} - 1110 p^{2} T^{10} + 994 p^{3} T^{11} + 293 p^{4} T^{12} + 96 p^{5} T^{13} - 7 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 9 T + 38 T^{2} - 85 T^{3} - 979 T^{4} - 85 p T^{5} + 38 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 19 T + 113 T^{2} + 86 T^{3} - 1117 T^{4} - 5456 T^{5} - 33160 T^{6} + 205495 T^{7} + 3154571 T^{8} + 205495 p T^{9} - 33160 p^{2} T^{10} - 5456 p^{3} T^{11} - 1117 p^{4} T^{12} + 86 p^{5} T^{13} + 113 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 6 T - 5 T^{2} + 15 T^{3} + 1295 T^{4} + 357 T^{5} - 47197 T^{6} + 151560 T^{7} + 294595 T^{8} + 151560 p T^{9} - 47197 p^{2} T^{10} + 357 p^{3} T^{11} + 1295 p^{4} T^{12} + 15 p^{5} T^{13} - 5 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 4 T - 97 T^{2} + 462 T^{3} + 2066 T^{4} - 23894 T^{5} + 174045 T^{6} + 437226 T^{7} - 12204441 T^{8} + 437226 p T^{9} + 174045 p^{2} T^{10} - 23894 p^{3} T^{11} + 2066 p^{4} T^{12} + 462 p^{5} T^{13} - 97 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 4 T - 85 T^{2} - 20 T^{3} + 2540 T^{4} + 13928 T^{5} + 163313 T^{6} - 521020 T^{7} - 12527665 T^{8} - 521020 p T^{9} + 163313 p^{2} T^{10} + 13928 p^{3} T^{11} + 2540 p^{4} T^{12} - 20 p^{5} T^{13} - 85 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 21 T + 293 T^{2} + 2900 T^{3} + 21441 T^{4} + 2900 p T^{5} + 293 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 4 T - 7 T^{2} - 23 T^{3} + 3951 T^{4} - 7279 T^{5} - 163935 T^{6} + 137206 T^{7} + 7103639 T^{8} + 137206 p T^{9} - 163935 p^{2} T^{10} - 7279 p^{3} T^{11} + 3951 p^{4} T^{12} - 23 p^{5} T^{13} - 7 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 3 T - 69 T^{2} + 487 T^{3} + 4629 T^{4} - 2812 T^{5} - 362068 T^{6} - 185784 T^{7} + 29291723 T^{8} - 185784 p T^{9} - 362068 p^{2} T^{10} - 2812 p^{3} T^{11} + 4629 p^{4} T^{12} + 487 p^{5} T^{13} - 69 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 19 T + 83 T^{2} + 161 T^{3} + 7463 T^{4} + 113974 T^{5} + 1180460 T^{6} + 5430220 T^{7} + 5701721 T^{8} + 5430220 p T^{9} + 1180460 p^{2} T^{10} + 113974 p^{3} T^{11} + 7463 p^{4} T^{12} + 161 p^{5} T^{13} + 83 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 2 T - 48 T^{2} + 78 T^{3} + 7033 T^{4} - 9842 T^{5} - 572520 T^{6} - 148460 T^{7} + 27924921 T^{8} - 148460 p T^{9} - 572520 p^{2} T^{10} - 9842 p^{3} T^{11} + 7033 p^{4} T^{12} + 78 p^{5} T^{13} - 48 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + T + 186 T^{2} - 37 T^{3} + 15845 T^{4} - 37 p T^{5} + 186 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 40 T + 727 T^{2} - 8050 T^{3} + 56358 T^{4} - 99810 T^{5} - 3707511 T^{6} + 61987150 T^{7} - 612781425 T^{8} + 61987150 p T^{9} - 3707511 p^{2} T^{10} - 99810 p^{3} T^{11} + 56358 p^{4} T^{12} - 8050 p^{5} T^{13} + 727 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 23 T + 222 T^{2} - 2419 T^{3} + 36136 T^{4} - 379918 T^{5} + 3242270 T^{6} - 31455482 T^{7} + 296606219 T^{8} - 31455482 p T^{9} + 3242270 p^{2} T^{10} - 379918 p^{3} T^{11} + 36136 p^{4} T^{12} - 2419 p^{5} T^{13} + 222 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 17 T - 4 T^{2} - 386 T^{3} + 11750 T^{4} + 154077 T^{5} + 1787170 T^{6} + 7920480 T^{7} - 39647009 T^{8} + 7920480 p T^{9} + 1787170 p^{2} T^{10} + 154077 p^{3} T^{11} + 11750 p^{4} T^{12} - 386 p^{5} T^{13} - 4 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 25 T + 105 T^{2} + 2035 T^{3} - 19109 T^{4} + 48470 T^{5} - 251250 T^{6} - 10913400 T^{7} + 221937861 T^{8} - 10913400 p T^{9} - 251250 p^{2} T^{10} + 48470 p^{3} T^{11} - 19109 p^{4} T^{12} + 2035 p^{5} T^{13} + 105 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 206 T^{2} - 400 T^{3} + 21551 T^{4} - 400 p T^{5} + 206 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 12 T - 15 T^{2} - 156 T^{3} + 22446 T^{4} - 71028 T^{5} - 2258399 T^{6} + 7047930 T^{7} + 179384859 T^{8} + 7047930 p T^{9} - 2258399 p^{2} T^{10} - 71028 p^{3} T^{11} + 22446 p^{4} T^{12} - 156 p^{5} T^{13} - 15 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.58052337342795661363555266480, −4.49720887092735302177943362325, −4.48866194315912875393263561267, −4.37860547133476898091053183192, −4.25257255028070642416632261537, −3.94409388030840584299724516983, −3.87158509454793883976978380897, −3.66520155658123024478327739909, −3.62922732704221439670024846826, −3.45944109648407894069149475793, −3.31409900325720910799422227417, −3.17736932505226846664499105911, −3.17098880500875079252696125832, −2.95004584178363077502492813901, −2.49407299857623984485928534609, −2.30927211238331005644916849345, −2.24527565528119289741118585536, −2.20579653561064809224983340824, −1.86728627734089396163499262235, −1.83702088180524767514776469097, −1.75295511597219884817295510815, −1.55954161134380505660863728422, −0.951042776947741349117279971711, −0.55790125013480933785255429363, −0.30505539636165100213954874438, 0.30505539636165100213954874438, 0.55790125013480933785255429363, 0.951042776947741349117279971711, 1.55954161134380505660863728422, 1.75295511597219884817295510815, 1.83702088180524767514776469097, 1.86728627734089396163499262235, 2.20579653561064809224983340824, 2.24527565528119289741118585536, 2.30927211238331005644916849345, 2.49407299857623984485928534609, 2.95004584178363077502492813901, 3.17098880500875079252696125832, 3.17736932505226846664499105911, 3.31409900325720910799422227417, 3.45944109648407894069149475793, 3.62922732704221439670024846826, 3.66520155658123024478327739909, 3.87158509454793883976978380897, 3.94409388030840584299724516983, 4.25257255028070642416632261537, 4.37860547133476898091053183192, 4.48866194315912875393263561267, 4.49720887092735302177943362325, 4.58052337342795661363555266480
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.