Dirichlet series
| L(s) = 1 | + 10·3-s − 4-s + 5·5-s − 10·7-s + 8-s + 55·9-s − 10·12-s − 6·13-s + 50·15-s + 16-s − 8·17-s − 5·20-s − 100·21-s + 10·24-s + 3·25-s + 220·27-s + 10·28-s + 14·29-s + 26·31-s + 6·32-s − 50·35-s − 55·36-s + 24·37-s − 60·39-s + 5·40-s − 19·41-s + 6·43-s + ⋯ |
| L(s) = 1 | + 5.77·3-s − 1/2·4-s + 2.23·5-s − 3.77·7-s + 0.353·8-s + 55/3·9-s − 2.88·12-s − 1.66·13-s + 12.9·15-s + 1/4·16-s − 1.94·17-s − 1.11·20-s − 21.8·21-s + 2.04·24-s + 3/5·25-s + 42.3·27-s + 1.88·28-s + 2.59·29-s + 4.66·31-s + 1.06·32-s − 8.45·35-s − 9.16·36-s + 3.94·37-s − 9.60·39-s + 0.790·40-s − 2.96·41-s + 0.914·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{10} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{10} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 7^{10} \cdot 11^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18254\times 10^{13}\) |
| Root analytic conductor: | \(4.50444\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 7^{10} \cdot 11^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(469.3337563\) |
| \(L(\frac12)\) | \(\approx\) | \(469.3337563\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{10} \) |
| 7 | \( ( 1 + T )^{10} \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + T^{2} - T^{3} - p^{3} T^{5} + p^{2} T^{6} - 7 T^{7} + 13 T^{8} - p^{3} T^{9} + 41 T^{10} - p^{4} T^{11} + 13 p^{2} T^{12} - 7 p^{3} T^{13} + p^{6} T^{14} - p^{8} T^{15} - p^{7} T^{17} + p^{8} T^{18} + p^{10} T^{20} \) |
| 5 | \( 1 - p T + 22 T^{2} - 46 T^{3} + 113 T^{4} - 108 T^{5} + 391 T^{6} - 538 T^{7} + 4002 T^{8} - 9087 T^{9} + 31806 T^{10} - 9087 p T^{11} + 4002 p^{2} T^{12} - 538 p^{3} T^{13} + 391 p^{4} T^{14} - 108 p^{5} T^{15} + 113 p^{6} T^{16} - 46 p^{7} T^{17} + 22 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 6 T + 5 p T^{2} + 368 T^{3} + 2385 T^{4} + 10918 T^{5} + 56756 T^{6} + 221994 T^{7} + 984630 T^{8} + 3474482 T^{9} + 13929014 T^{10} + 3474482 p T^{11} + 984630 p^{2} T^{12} + 221994 p^{3} T^{13} + 56756 p^{4} T^{14} + 10918 p^{5} T^{15} + 2385 p^{6} T^{16} + 368 p^{7} T^{17} + 5 p^{9} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 8 T + 8 p T^{2} + 954 T^{3} + 9093 T^{4} + 54096 T^{5} + 378365 T^{6} + 1910556 T^{7} + 10713786 T^{8} + 46048194 T^{9} + 215035238 T^{10} + 46048194 p T^{11} + 10713786 p^{2} T^{12} + 1910556 p^{3} T^{13} + 378365 p^{4} T^{14} + 54096 p^{5} T^{15} + 9093 p^{6} T^{16} + 954 p^{7} T^{17} + 8 p^{9} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 69 T^{2} - 136 T^{3} + 2758 T^{4} - 8362 T^{5} + 89118 T^{6} - 289638 T^{7} + 2274617 T^{8} - 397140 p T^{9} + 46843450 T^{10} - 397140 p^{2} T^{11} + 2274617 p^{2} T^{12} - 289638 p^{3} T^{13} + 89118 p^{4} T^{14} - 8362 p^{5} T^{15} + 2758 p^{6} T^{16} - 136 p^{7} T^{17} + 69 p^{8} T^{18} + p^{10} T^{20} \) | |
| 23 | \( 1 + 124 T^{2} + 38 T^{3} + 8330 T^{4} + 158 p T^{5} + 381506 T^{6} + 180342 T^{7} + 12965000 T^{8} + 5936040 T^{9} + 338330610 T^{10} + 5936040 p T^{11} + 12965000 p^{2} T^{12} + 180342 p^{3} T^{13} + 381506 p^{4} T^{14} + 158 p^{6} T^{15} + 8330 p^{6} T^{16} + 38 p^{7} T^{17} + 124 p^{8} T^{18} + p^{10} T^{20} \) | |
| 29 | \( 1 - 14 T + 200 T^{2} - 1928 T^{3} + 18575 T^{4} - 144068 T^{5} + 1110481 T^{6} - 7337936 T^{7} + 48191364 T^{8} - 277661302 T^{9} + 1587859174 T^{10} - 277661302 p T^{11} + 48191364 p^{2} T^{12} - 7337936 p^{3} T^{13} + 1110481 p^{4} T^{14} - 144068 p^{5} T^{15} + 18575 p^{6} T^{16} - 1928 p^{7} T^{17} + 200 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 26 T + 482 T^{2} - 6400 T^{3} + 71453 T^{4} - 670846 T^{5} + 5624329 T^{6} - 41976980 T^{7} + 287918626 T^{8} - 1800633596 T^{9} + 10465823578 T^{10} - 1800633596 p T^{11} + 287918626 p^{2} T^{12} - 41976980 p^{3} T^{13} + 5624329 p^{4} T^{14} - 670846 p^{5} T^{15} + 71453 p^{6} T^{16} - 6400 p^{7} T^{17} + 482 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 24 T + 440 T^{2} - 5840 T^{3} + 65456 T^{4} - 634664 T^{5} + 5522032 T^{6} - 43933344 T^{7} + 323265704 T^{8} - 2198445368 T^{9} + 13921174566 T^{10} - 2198445368 p T^{11} + 323265704 p^{2} T^{12} - 43933344 p^{3} T^{13} + 5522032 p^{4} T^{14} - 634664 p^{5} T^{15} + 65456 p^{6} T^{16} - 5840 p^{7} T^{17} + 440 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 19 T + 484 T^{2} + 6388 T^{3} + 94475 T^{4} + 963532 T^{5} + 10457893 T^{6} + 86623352 T^{7} + 748989936 T^{8} + 5154844493 T^{9} + 36776727094 T^{10} + 5154844493 p T^{11} + 748989936 p^{2} T^{12} + 86623352 p^{3} T^{13} + 10457893 p^{4} T^{14} + 963532 p^{5} T^{15} + 94475 p^{6} T^{16} + 6388 p^{7} T^{17} + 484 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 6 T + 312 T^{2} - 1828 T^{3} + 47319 T^{4} - 259736 T^{5} + 4576717 T^{6} - 22838020 T^{7} + 311402464 T^{8} - 1377461130 T^{9} + 15540959670 T^{10} - 1377461130 p T^{11} + 311402464 p^{2} T^{12} - 22838020 p^{3} T^{13} + 4576717 p^{4} T^{14} - 259736 p^{5} T^{15} + 47319 p^{6} T^{16} - 1828 p^{7} T^{17} + 312 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 15 T + 333 T^{2} - 4161 T^{3} + 56109 T^{4} - 576028 T^{5} + 127236 p T^{6} - 51937604 T^{7} + 445599026 T^{8} - 3323250890 T^{9} + 24323199470 T^{10} - 3323250890 p T^{11} + 445599026 p^{2} T^{12} - 51937604 p^{3} T^{13} + 127236 p^{5} T^{14} - 576028 p^{5} T^{15} + 56109 p^{6} T^{16} - 4161 p^{7} T^{17} + 333 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + T + 254 T^{2} - 378 T^{3} + 28817 T^{4} - 120372 T^{5} + 2050339 T^{6} - 14991722 T^{7} + 109654718 T^{8} - 1160429241 T^{9} + 5512758766 T^{10} - 1160429241 p T^{11} + 109654718 p^{2} T^{12} - 14991722 p^{3} T^{13} + 2050339 p^{4} T^{14} - 120372 p^{5} T^{15} + 28817 p^{6} T^{16} - 378 p^{7} T^{17} + 254 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 23 T + 495 T^{2} - 6037 T^{3} + 71081 T^{4} - 556772 T^{5} + 4700924 T^{6} - 26168124 T^{7} + 209482574 T^{8} - 1016636482 T^{9} + 10462754634 T^{10} - 1016636482 p T^{11} + 209482574 p^{2} T^{12} - 26168124 p^{3} T^{13} + 4700924 p^{4} T^{14} - 556772 p^{5} T^{15} + 71081 p^{6} T^{16} - 6037 p^{7} T^{17} + 495 p^{8} T^{18} - 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 215 T^{2} + 490 T^{3} + 23725 T^{4} + 88654 T^{5} + 1906100 T^{6} + 8267970 T^{7} + 125871770 T^{8} + 598496950 T^{9} + 7588984906 T^{10} + 598496950 p T^{11} + 125871770 p^{2} T^{12} + 8267970 p^{3} T^{13} + 1906100 p^{4} T^{14} + 88654 p^{5} T^{15} + 23725 p^{6} T^{16} + 490 p^{7} T^{17} + 215 p^{8} T^{18} + p^{10} T^{20} \) | |
| 67 | \( 1 - 38 T + 939 T^{2} - 16920 T^{3} + 253022 T^{4} - 3231648 T^{5} + 37034650 T^{6} - 384823180 T^{7} + 3720320353 T^{8} - 33446868814 T^{9} + 283134556406 T^{10} - 33446868814 p T^{11} + 3720320353 p^{2} T^{12} - 384823180 p^{3} T^{13} + 37034650 p^{4} T^{14} - 3231648 p^{5} T^{15} + 253022 p^{6} T^{16} - 16920 p^{7} T^{17} + 939 p^{8} T^{18} - 38 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 26 T + 632 T^{2} - 9130 T^{3} + 126663 T^{4} - 1295526 T^{5} + 13933979 T^{6} - 123894450 T^{7} + 1257395276 T^{8} - 10721089596 T^{9} + 100241082738 T^{10} - 10721089596 p T^{11} + 1257395276 p^{2} T^{12} - 123894450 p^{3} T^{13} + 13933979 p^{4} T^{14} - 1295526 p^{5} T^{15} + 126663 p^{6} T^{16} - 9130 p^{7} T^{17} + 632 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - T + 393 T^{2} - 817 T^{3} + 74513 T^{4} - 250396 T^{5} + 9185572 T^{6} - 43089004 T^{7} + 854063166 T^{8} - 4726643438 T^{9} + 66584553318 T^{10} - 4726643438 p T^{11} + 854063166 p^{2} T^{12} - 43089004 p^{3} T^{13} + 9185572 p^{4} T^{14} - 250396 p^{5} T^{15} + 74513 p^{6} T^{16} - 817 p^{7} T^{17} + 393 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 5 T + 552 T^{2} + 2604 T^{3} + 151027 T^{4} + 658718 T^{5} + 26569829 T^{6} + 105054020 T^{7} + 3304178864 T^{8} + 11590016269 T^{9} + 302495065710 T^{10} + 11590016269 p T^{11} + 3304178864 p^{2} T^{12} + 105054020 p^{3} T^{13} + 26569829 p^{4} T^{14} + 658718 p^{5} T^{15} + 151027 p^{6} T^{16} + 2604 p^{7} T^{17} + 552 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 6 T + 490 T^{2} + 3578 T^{3} + 118805 T^{4} + 976848 T^{5} + 19084536 T^{6} + 164833504 T^{7} + 2271305170 T^{8} + 19148739052 T^{9} + 211320377724 T^{10} + 19148739052 p T^{11} + 2271305170 p^{2} T^{12} + 164833504 p^{3} T^{13} + 19084536 p^{4} T^{14} + 976848 p^{5} T^{15} + 118805 p^{6} T^{16} + 3578 p^{7} T^{17} + 490 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 9 T + 640 T^{2} + 4050 T^{3} + 178355 T^{4} + 698876 T^{5} + 29313265 T^{6} + 51732294 T^{7} + 3369419368 T^{8} + 1036172651 T^{9} + 317703552166 T^{10} + 1036172651 p T^{11} + 3369419368 p^{2} T^{12} + 51732294 p^{3} T^{13} + 29313265 p^{4} T^{14} + 698876 p^{5} T^{15} + 178355 p^{6} T^{16} + 4050 p^{7} T^{17} + 640 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 24 T + 1069 T^{2} - 19604 T^{3} + 490777 T^{4} - 7233428 T^{5} + 130302052 T^{6} - 1582958620 T^{7} + 22393814246 T^{8} - 226465922980 T^{9} + 2616113341790 T^{10} - 226465922980 p T^{11} + 22393814246 p^{2} T^{12} - 1582958620 p^{3} T^{13} + 130302052 p^{4} T^{14} - 7233428 p^{5} T^{15} + 490777 p^{6} T^{16} - 19604 p^{7} T^{17} + 1069 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.08666034752823772739176060290, −2.96001334720551070057711812992, −2.84674637708011702793350720291, −2.66760333215944080496524066908, −2.62934140540781408489038831868, −2.62134639770921971849914064134, −2.61196705185728362051280744121, −2.59249928589679999508846629403, −2.54772793470308109222937591042, −2.27977473963659989488322437907, −2.23181761229830217178621450528, −1.98846473844422060632767423234, −1.98361920140482984302870388501, −1.95578860605304668310266125257, −1.91647824789264311802059690484, −1.74897905188069208488584345920, −1.52693155564475760941363963217, −1.32630170646631392131259301264, −1.09006601728989201404024630742, −0.934070198580870349618378020094, −0.859212824965582697741151841781, −0.78817511761074487140637044959, −0.69632718891430904561529425068, −0.55889117766326179282198226644, −0.33320481613215275451844342677, 0.33320481613215275451844342677, 0.55889117766326179282198226644, 0.69632718891430904561529425068, 0.78817511761074487140637044959, 0.859212824965582697741151841781, 0.934070198580870349618378020094, 1.09006601728989201404024630742, 1.32630170646631392131259301264, 1.52693155564475760941363963217, 1.74897905188069208488584345920, 1.91647824789264311802059690484, 1.95578860605304668310266125257, 1.98361920140482984302870388501, 1.98846473844422060632767423234, 2.23181761229830217178621450528, 2.27977473963659989488322437907, 2.54772793470308109222937591042, 2.59249928589679999508846629403, 2.61196705185728362051280744121, 2.62134639770921971849914064134, 2.62934140540781408489038831868, 2.66760333215944080496524066908, 2.84674637708011702793350720291, 2.96001334720551070057711812992, 3.08666034752823772739176060290
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.