Dirichlet series
| L(s) = 1 | + 2-s − 4·5-s − 7·7-s − 8·9-s − 4·10-s − 2·11-s + 2·13-s − 7·14-s − 9·17-s − 8·18-s + 2·19-s − 2·22-s + 21·23-s + 5·25-s + 2·26-s − 2·29-s + 11·31-s − 9·34-s + 28·35-s − 18·37-s + 2·38-s + 5·41-s − 21·43-s + 32·45-s + 21·46-s − 22·47-s + 40·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.78·5-s − 2.64·7-s − 8/3·9-s − 1.26·10-s − 0.603·11-s + 0.554·13-s − 1.87·14-s − 2.18·17-s − 1.88·18-s + 0.458·19-s − 0.426·22-s + 4.37·23-s + 25-s + 0.392·26-s − 0.371·29-s + 1.97·31-s − 1.54·34-s + 4.73·35-s − 2.95·37-s + 0.324·38-s + 0.780·41-s − 3.20·43-s + 4.77·45-s + 3.09·46-s − 3.20·47-s + 40/7·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.47043\times 10^{-5}\) |
| Root analytic conductor: | \(0.606062\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1007089039\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1007089039\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 23 | \( 1 - 21 T + 210 T^{2} - 1528 T^{3} + 9758 T^{4} - 52491 T^{5} + 9758 p T^{6} - 1528 p^{2} T^{7} + 210 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 3 | \( 1 + 8 T^{2} + 31 T^{4} + 50 T^{6} - 11 T^{7} - 106 T^{8} - 77 T^{9} - 683 T^{10} - 77 p T^{11} - 106 p^{2} T^{12} - 11 p^{3} T^{13} + 50 p^{4} T^{14} + 31 p^{6} T^{16} + 8 p^{8} T^{18} + p^{10} T^{20} \) |
| 5 | \( 1 + 4 T + 11 T^{2} + 13 T^{3} + 52 T^{4} + 121 T^{5} + 16 p^{2} T^{6} + 577 T^{7} + 1969 T^{8} + 2373 T^{9} + 8051 T^{10} + 2373 p T^{11} + 1969 p^{2} T^{12} + 577 p^{3} T^{13} + 16 p^{6} T^{14} + 121 p^{5} T^{15} + 52 p^{6} T^{16} + 13 p^{7} T^{17} + 11 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 + p T + 9 T^{2} - 9 p T^{3} - 317 T^{4} - 67 p T^{5} + 1268 T^{6} + 7451 T^{7} + 11722 T^{8} - 24344 T^{9} - 147489 T^{10} - 24344 p T^{11} + 11722 p^{2} T^{12} + 7451 p^{3} T^{13} + 1268 p^{4} T^{14} - 67 p^{6} T^{15} - 317 p^{6} T^{16} - 9 p^{8} T^{17} + 9 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 2 T + 15 T^{2} + 30 T^{3} + 269 T^{4} + 527 T^{5} + 2165 T^{6} + 3978 T^{7} + 33311 T^{8} + 37131 T^{9} + 279093 T^{10} + 37131 p T^{11} + 33311 p^{2} T^{12} + 3978 p^{3} T^{13} + 2165 p^{4} T^{14} + 527 p^{5} T^{15} + 269 p^{6} T^{16} + 30 p^{7} T^{17} + 15 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 2 T + 35 T^{2} + 22 T^{3} + 623 T^{4} + 1152 T^{5} + 13577 T^{6} + 18700 T^{7} + 232853 T^{8} + 336810 T^{9} + 3188371 T^{10} + 336810 p T^{11} + 232853 p^{2} T^{12} + 18700 p^{3} T^{13} + 13577 p^{4} T^{14} + 1152 p^{5} T^{15} + 623 p^{6} T^{16} + 22 p^{7} T^{17} + 35 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 9 T + 64 T^{2} + 335 T^{3} + 1619 T^{4} + 7666 T^{5} + 33045 T^{6} + 153971 T^{7} + 662758 T^{8} + 2706609 T^{9} + 10980199 T^{10} + 2706609 p T^{11} + 662758 p^{2} T^{12} + 153971 p^{3} T^{13} + 33045 p^{4} T^{14} + 7666 p^{5} T^{15} + 1619 p^{6} T^{16} + 335 p^{7} T^{17} + 64 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 2 T - 4 T^{2} - 196 T^{3} - 203 T^{4} - 446 T^{5} + 30500 T^{6} + 63645 T^{7} + 291196 T^{8} - 2476969 T^{9} - 7759399 T^{10} - 2476969 p T^{11} + 291196 p^{2} T^{12} + 63645 p^{3} T^{13} + 30500 p^{4} T^{14} - 446 p^{5} T^{15} - 203 p^{6} T^{16} - 196 p^{7} T^{17} - 4 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 2 T + 19 T^{2} + 178 T^{3} + 1708 T^{4} + 10926 T^{5} + 66282 T^{6} + 362190 T^{7} + 1989298 T^{8} + 15206180 T^{9} + 74728193 T^{10} + 15206180 p T^{11} + 1989298 p^{2} T^{12} + 362190 p^{3} T^{13} + 66282 p^{4} T^{14} + 10926 p^{5} T^{15} + 1708 p^{6} T^{16} + 178 p^{7} T^{17} + 19 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 11 T + 79 T^{2} - 264 T^{3} - 51 T^{4} + 11033 T^{5} - 2424 p T^{6} + 293667 T^{7} + 690662 T^{8} - 11513337 T^{9} + 91215431 T^{10} - 11513337 p T^{11} + 690662 p^{2} T^{12} + 293667 p^{3} T^{13} - 2424 p^{5} T^{14} + 11033 p^{5} T^{15} - 51 p^{6} T^{16} - 264 p^{7} T^{17} + 79 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 18 T + 188 T^{2} + 1354 T^{3} + 8374 T^{4} + 62310 T^{5} + 523707 T^{6} + 4137902 T^{7} + 26154892 T^{8} + 137331222 T^{9} + 755646705 T^{10} + 137331222 p T^{11} + 26154892 p^{2} T^{12} + 4137902 p^{3} T^{13} + 523707 p^{4} T^{14} + 62310 p^{5} T^{15} + 8374 p^{6} T^{16} + 1354 p^{7} T^{17} + 188 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 5 T - 16 T^{2} - 111 T^{3} + 1343 T^{4} + 12796 T^{5} - 17183 T^{6} - 222351 T^{7} - 4440718 T^{8} + 13261721 T^{9} + 181368727 T^{10} + 13261721 p T^{11} - 4440718 p^{2} T^{12} - 222351 p^{3} T^{13} - 17183 p^{4} T^{14} + 12796 p^{5} T^{15} + 1343 p^{6} T^{16} - 111 p^{7} T^{17} - 16 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 21 T + 189 T^{2} + 1339 T^{3} + 10400 T^{4} + 53771 T^{5} + 84130 T^{6} - 809225 T^{7} - 14823115 T^{8} - 171784492 T^{9} - 1343910083 T^{10} - 171784492 p T^{11} - 14823115 p^{2} T^{12} - 809225 p^{3} T^{13} + 84130 p^{4} T^{14} + 53771 p^{5} T^{15} + 10400 p^{6} T^{16} + 1339 p^{7} T^{17} + 189 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 11 T + 169 T^{2} + 1342 T^{3} + 13169 T^{4} + 81433 T^{5} + 13169 p T^{6} + 1342 p^{2} T^{7} + 169 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 7 T - 37 T^{2} - 630 T^{3} + 2303 T^{4} + 61985 T^{5} + 215608 T^{6} - 2997609 T^{7} - 26274170 T^{8} + 59400163 T^{9} + 1783141799 T^{10} + 59400163 p T^{11} - 26274170 p^{2} T^{12} - 2997609 p^{3} T^{13} + 215608 p^{4} T^{14} + 61985 p^{5} T^{15} + 2303 p^{6} T^{16} - 630 p^{7} T^{17} - 37 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 43 T + 800 T^{2} - 7443 T^{3} + 16835 T^{4} + 431794 T^{5} - 5767595 T^{6} + 30227089 T^{7} + 49048862 T^{8} - 2392252041 T^{9} + 24409976821 T^{10} - 2392252041 p T^{11} + 49048862 p^{2} T^{12} + 30227089 p^{3} T^{13} - 5767595 p^{4} T^{14} + 431794 p^{5} T^{15} + 16835 p^{6} T^{16} - 7443 p^{7} T^{17} + 800 p^{8} T^{18} - 43 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 3 T - 8 T^{2} + 409 T^{3} + 4971 T^{4} + 18146 T^{5} - 4879 T^{6} - 91657 T^{7} + 10195096 T^{8} + 35966089 T^{9} - 996076401 T^{10} + 35966089 p T^{11} + 10195096 p^{2} T^{12} - 91657 p^{3} T^{13} - 4879 p^{4} T^{14} + 18146 p^{5} T^{15} + 4971 p^{6} T^{16} + 409 p^{7} T^{17} - 8 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + T - 121 T^{2} - 364 T^{3} - 1453 T^{4} + 18447 T^{5} + 633502 T^{6} + 3263107 T^{7} + 10635966 T^{8} - 210183997 T^{9} - 3988669301 T^{10} - 210183997 p T^{11} + 10635966 p^{2} T^{12} + 3263107 p^{3} T^{13} + 633502 p^{4} T^{14} + 18447 p^{5} T^{15} - 1453 p^{6} T^{16} - 364 p^{7} T^{17} - 121 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 11 T + 39 T^{2} - 616 T^{3} - 6091 T^{4} - 104313 T^{5} - 708624 T^{6} - 2547787 T^{7} + 45967942 T^{8} + 367259057 T^{9} + 4248223671 T^{10} + 367259057 p T^{11} + 45967942 p^{2} T^{12} - 2547787 p^{3} T^{13} - 708624 p^{4} T^{14} - 104313 p^{5} T^{15} - 6091 p^{6} T^{16} - 616 p^{7} T^{17} + 39 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 28 T + 293 T^{2} + 869 T^{3} + 1150 T^{4} + 210587 T^{5} + 3515862 T^{6} + 23321309 T^{7} + 103858593 T^{8} + 1356764141 T^{9} + 16809322991 T^{10} + 1356764141 p T^{11} + 103858593 p^{2} T^{12} + 23321309 p^{3} T^{13} + 3515862 p^{4} T^{14} + 210587 p^{5} T^{15} + 1150 p^{6} T^{16} + 869 p^{7} T^{17} + 293 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 34 T + 824 T^{2} - 16189 T^{3} + 266903 T^{4} - 3896876 T^{5} + 50702252 T^{6} - 600841755 T^{7} + 6526316054 T^{8} - 65186046398 T^{9} + 602818153939 T^{10} - 65186046398 p T^{11} + 6526316054 p^{2} T^{12} - 600841755 p^{3} T^{13} + 50702252 p^{4} T^{14} - 3896876 p^{5} T^{15} + 266903 p^{6} T^{16} - 16189 p^{7} T^{17} + 824 p^{8} T^{18} - 34 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 3 T - 360 T^{2} - 801 T^{3} + 69409 T^{4} + 98270 T^{5} - 9578553 T^{6} - 6511631 T^{7} + 1059736710 T^{8} + 194052655 T^{9} - 96570716771 T^{10} + 194052655 p T^{11} + 1059736710 p^{2} T^{12} - 6511631 p^{3} T^{13} - 9578553 p^{4} T^{14} + 98270 p^{5} T^{15} + 69409 p^{6} T^{16} - 801 p^{7} T^{17} - 360 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 49 T + 1102 T^{2} + 159 p T^{3} + 88001 T^{4} - 422686 T^{5} - 15999347 T^{6} - 162304917 T^{7} - 334444508 T^{8} + 12801982375 T^{9} + 191555953423 T^{10} + 12801982375 p T^{11} - 334444508 p^{2} T^{12} - 162304917 p^{3} T^{13} - 15999347 p^{4} T^{14} - 422686 p^{5} T^{15} + 88001 p^{6} T^{16} + 159 p^{8} T^{17} + 1102 p^{8} T^{18} + 49 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 16 T - 193 T^{2} + 4552 T^{3} + 6103 T^{4} - 576361 T^{5} + 2094641 T^{6} + 37046928 T^{7} - 323613245 T^{8} - 1067492705 T^{9} + 32351755007 T^{10} - 1067492705 p T^{11} - 323613245 p^{2} T^{12} + 37046928 p^{3} T^{13} + 2094641 p^{4} T^{14} - 576361 p^{5} T^{15} + 6103 p^{6} T^{16} + 4552 p^{7} T^{17} - 193 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−6.85735763705587468664019941409, −6.75725296116448395531755582806, −6.56113108777012281603866638524, −6.52118775130167478013110048034, −6.43822082882635961482822196777, −5.96898462781021610971408041398, −5.68087471540256896321167036375, −5.68029211454484492482606442953, −5.67437993554358036156859774183, −5.56430799547278332103524847504, −5.06548363283381029916753139471, −4.96567925681894845956541314273, −4.80415990622208720577708199194, −4.65812883711541595631540595394, −4.48916111198412176753861727355, −4.42509720885890363764982916985, −3.69290666418932711763248501216, −3.66207636302238012715618758039, −3.54842469103749240811478885644, −3.40300621886023635503152189152, −3.17671238151809933240179534843, −3.03495215890506590148935069652, −2.69664575449633268986665157642, −2.61491187203210427702432454863, −2.08715073228369810181640206899, 2.08715073228369810181640206899, 2.61491187203210427702432454863, 2.69664575449633268986665157642, 3.03495215890506590148935069652, 3.17671238151809933240179534843, 3.40300621886023635503152189152, 3.54842469103749240811478885644, 3.66207636302238012715618758039, 3.69290666418932711763248501216, 4.42509720885890363764982916985, 4.48916111198412176753861727355, 4.65812883711541595631540595394, 4.80415990622208720577708199194, 4.96567925681894845956541314273, 5.06548363283381029916753139471, 5.56430799547278332103524847504, 5.67437993554358036156859774183, 5.68029211454484492482606442953, 5.68087471540256896321167036375, 5.96898462781021610971408041398, 6.43822082882635961482822196777, 6.52118775130167478013110048034, 6.56113108777012281603866638524, 6.75725296116448395531755582806, 6.85735763705587468664019941409
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.