Dirichlet series
| L(s) = 1 | − 10·3-s − 10·5-s + 7-s + 55·9-s − 11-s − 6·13-s + 100·15-s − 9·17-s + 2·19-s − 10·21-s + 7·23-s + 31·25-s − 220·27-s − 13·29-s + 23·31-s + 10·33-s − 10·35-s − 6·37-s + 60·39-s − 12·41-s − 550·45-s + 10·47-s − 31·49-s + 90·51-s − 26·53-s + 10·55-s − 20·57-s + ⋯ |
| L(s) = 1 | − 5.77·3-s − 4.47·5-s + 0.377·7-s + 55/3·9-s − 0.301·11-s − 1.66·13-s + 25.8·15-s − 2.18·17-s + 0.458·19-s − 2.18·21-s + 1.45·23-s + 31/5·25-s − 42.3·27-s − 2.41·29-s + 4.13·31-s + 1.74·33-s − 1.69·35-s − 0.986·37-s + 9.60·39-s − 1.87·41-s − 81.9·45-s + 1.45·47-s − 4.42·49-s + 12.6·51-s − 3.57·53-s + 1.34·55-s − 2.64·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{30} \cdot 3^{10} \cdot 167^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.12732\times 10^{15}\) |
| Root analytic conductor: | \(5.65721\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 2^{30} \cdot 3^{10} \cdot 167^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
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 | \( 1 \) |
| 3 | \( ( 1 + T )^{10} \) | |
| 167 | \( ( 1 + T )^{10} \) | |
| good | 5 | \( 1 + 2 p T + 69 T^{2} + 73 p T^{3} + 1608 T^{4} + 6162 T^{5} + 836 p^{2} T^{6} + 63916 T^{7} + 177614 T^{8} + 451353 T^{9} + 1053362 T^{10} + 451353 p T^{11} + 177614 p^{2} T^{12} + 63916 p^{3} T^{13} + 836 p^{6} T^{14} + 6162 p^{5} T^{15} + 1608 p^{6} T^{16} + 73 p^{8} T^{17} + 69 p^{8} T^{18} + 2 p^{10} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - T + 32 T^{2} - 38 T^{3} + 544 T^{4} - 725 T^{5} + 6513 T^{6} - 9463 T^{7} + 61102 T^{8} - 89363 T^{9} + 470216 T^{10} - 89363 p T^{11} + 61102 p^{2} T^{12} - 9463 p^{3} T^{13} + 6513 p^{4} T^{14} - 725 p^{5} T^{15} + 544 p^{6} T^{16} - 38 p^{7} T^{17} + 32 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + T + 57 T^{2} + 23 T^{3} + 142 p T^{4} - 101 T^{5} + 28412 T^{6} - 11647 T^{7} + 399505 T^{8} - 230292 T^{9} + 4729134 T^{10} - 230292 p T^{11} + 399505 p^{2} T^{12} - 11647 p^{3} T^{13} + 28412 p^{4} T^{14} - 101 p^{5} T^{15} + 142 p^{7} T^{16} + 23 p^{7} T^{17} + 57 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 6 T + 83 T^{2} + 478 T^{3} + 3637 T^{4} + 18563 T^{5} + 104668 T^{6} + 465567 T^{7} + 2135350 T^{8} + 8277926 T^{9} + 32123658 T^{10} + 8277926 p T^{11} + 2135350 p^{2} T^{12} + 465567 p^{3} T^{13} + 104668 p^{4} T^{14} + 18563 p^{5} T^{15} + 3637 p^{6} T^{16} + 478 p^{7} T^{17} + 83 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 9 T + 118 T^{2} + 837 T^{3} + 6250 T^{4} + 35453 T^{5} + 196924 T^{6} + 935647 T^{7} + 4342997 T^{8} + 18507198 T^{9} + 78377748 T^{10} + 18507198 p T^{11} + 4342997 p^{2} T^{12} + 935647 p^{3} T^{13} + 196924 p^{4} T^{14} + 35453 p^{5} T^{15} + 6250 p^{6} T^{16} + 837 p^{7} T^{17} + 118 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 2 T + 85 T^{2} - 2 T^{3} + 3473 T^{4} + 5923 T^{5} + 99516 T^{6} + 314949 T^{7} + 2275338 T^{8} + 9282398 T^{9} + 45237014 T^{10} + 9282398 p T^{11} + 2275338 p^{2} T^{12} + 314949 p^{3} T^{13} + 99516 p^{4} T^{14} + 5923 p^{5} T^{15} + 3473 p^{6} T^{16} - 2 p^{7} T^{17} + 85 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 7 T + 95 T^{2} - 569 T^{3} + 5344 T^{4} - 29007 T^{5} + 221988 T^{6} - 1080533 T^{7} + 7067347 T^{8} - 31317240 T^{9} + 181558482 T^{10} - 31317240 p T^{11} + 7067347 p^{2} T^{12} - 1080533 p^{3} T^{13} + 221988 p^{4} T^{14} - 29007 p^{5} T^{15} + 5344 p^{6} T^{16} - 569 p^{7} T^{17} + 95 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 13 T + 159 T^{2} + 939 T^{3} + 6582 T^{4} + 25017 T^{5} + 196396 T^{6} + 616513 T^{7} + 4522549 T^{8} + 29470 T^{9} + 53516466 T^{10} + 29470 p T^{11} + 4522549 p^{2} T^{12} + 616513 p^{3} T^{13} + 196396 p^{4} T^{14} + 25017 p^{5} T^{15} + 6582 p^{6} T^{16} + 939 p^{7} T^{17} + 159 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 23 T + 381 T^{2} - 4487 T^{3} + 1424 p T^{4} - 360340 T^{5} + 2585914 T^{6} - 16406937 T^{7} + 96372624 T^{8} - 537949941 T^{9} + 2990336088 T^{10} - 537949941 p T^{11} + 96372624 p^{2} T^{12} - 16406937 p^{3} T^{13} + 2585914 p^{4} T^{14} - 360340 p^{5} T^{15} + 1424 p^{7} T^{16} - 4487 p^{7} T^{17} + 381 p^{8} T^{18} - 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 6 T + 193 T^{2} + 441 T^{3} + 14860 T^{4} - 11088 T^{5} + 798266 T^{6} - 1849094 T^{7} + 42290634 T^{8} - 74596125 T^{9} + 1868536464 T^{10} - 74596125 p T^{11} + 42290634 p^{2} T^{12} - 1849094 p^{3} T^{13} + 798266 p^{4} T^{14} - 11088 p^{5} T^{15} + 14860 p^{6} T^{16} + 441 p^{7} T^{17} + 193 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 12 T + 294 T^{2} + 3115 T^{3} + 43285 T^{4} + 398010 T^{5} + 4063184 T^{6} + 32482454 T^{7} + 266719470 T^{8} + 1847808865 T^{9} + 12746910316 T^{10} + 1847808865 p T^{11} + 266719470 p^{2} T^{12} + 32482454 p^{3} T^{13} + 4063184 p^{4} T^{14} + 398010 p^{5} T^{15} + 43285 p^{6} T^{16} + 3115 p^{7} T^{17} + 294 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 252 T^{2} - 279 T^{3} + 32851 T^{4} - 55784 T^{5} + 2880724 T^{6} - 5482364 T^{7} + 186179700 T^{8} - 341733543 T^{9} + 9146061736 T^{10} - 341733543 p T^{11} + 186179700 p^{2} T^{12} - 5482364 p^{3} T^{13} + 2880724 p^{4} T^{14} - 55784 p^{5} T^{15} + 32851 p^{6} T^{16} - 279 p^{7} T^{17} + 252 p^{8} T^{18} + p^{10} T^{20} \) | |
| 47 | \( 1 - 10 T + 441 T^{2} - 3860 T^{3} + 88852 T^{4} - 677217 T^{5} + 10763404 T^{6} - 70906584 T^{7} + 867634490 T^{8} - 4884101757 T^{9} + 48609497732 T^{10} - 4884101757 p T^{11} + 867634490 p^{2} T^{12} - 70906584 p^{3} T^{13} + 10763404 p^{4} T^{14} - 677217 p^{5} T^{15} + 88852 p^{6} T^{16} - 3860 p^{7} T^{17} + 441 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 26 T + 577 T^{2} + 8762 T^{3} + 121850 T^{4} + 1386171 T^{5} + 14819742 T^{6} + 137932446 T^{7} + 1227120420 T^{8} + 9761201687 T^{9} + 74837899402 T^{10} + 9761201687 p T^{11} + 1227120420 p^{2} T^{12} + 137932446 p^{3} T^{13} + 14819742 p^{4} T^{14} + 1386171 p^{5} T^{15} + 121850 p^{6} T^{16} + 8762 p^{7} T^{17} + 577 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 10 T + 7 p T^{2} + 3896 T^{3} + 85628 T^{4} + 727669 T^{5} + 11417612 T^{6} + 85854922 T^{7} + 1070499278 T^{8} + 7042654803 T^{9} + 73409697452 T^{10} + 7042654803 p T^{11} + 1070499278 p^{2} T^{12} + 85854922 p^{3} T^{13} + 11417612 p^{4} T^{14} + 727669 p^{5} T^{15} + 85628 p^{6} T^{16} + 3896 p^{7} T^{17} + 7 p^{9} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 10 T + 531 T^{2} + 4616 T^{3} + 130505 T^{4} + 988345 T^{5} + 19590364 T^{6} + 128900293 T^{7} + 1991769246 T^{8} + 11288878956 T^{9} + 143433528194 T^{10} + 11288878956 p T^{11} + 1991769246 p^{2} T^{12} + 128900293 p^{3} T^{13} + 19590364 p^{4} T^{14} + 988345 p^{5} T^{15} + 130505 p^{6} T^{16} + 4616 p^{7} T^{17} + 531 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 5 T + 356 T^{2} + 1893 T^{3} + 65824 T^{4} + 371066 T^{5} + 8314295 T^{6} + 48002653 T^{7} + 791130118 T^{8} + 4397053995 T^{9} + 59302045306 T^{10} + 4397053995 p T^{11} + 791130118 p^{2} T^{12} + 48002653 p^{3} T^{13} + 8314295 p^{4} T^{14} + 371066 p^{5} T^{15} + 65824 p^{6} T^{16} + 1893 p^{7} T^{17} + 356 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 25 T + 758 T^{2} - 12926 T^{3} + 235700 T^{4} - 3124326 T^{5} + 42759652 T^{6} - 464718772 T^{7} + 5145626719 T^{8} - 46901534195 T^{9} + 433380203268 T^{10} - 46901534195 p T^{11} + 5145626719 p^{2} T^{12} - 464718772 p^{3} T^{13} + 42759652 p^{4} T^{14} - 3124326 p^{5} T^{15} + 235700 p^{6} T^{16} - 12926 p^{7} T^{17} + 758 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 8 T + 557 T^{2} + 4242 T^{3} + 145301 T^{4} + 1047687 T^{5} + 23623980 T^{6} + 158940879 T^{7} + 2687146702 T^{8} + 16400535416 T^{9} + 226096563398 T^{10} + 16400535416 p T^{11} + 2687146702 p^{2} T^{12} + 158940879 p^{3} T^{13} + 23623980 p^{4} T^{14} + 1047687 p^{5} T^{15} + 145301 p^{6} T^{16} + 4242 p^{7} T^{17} + 557 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 26 T + 782 T^{2} - 14711 T^{3} + 270063 T^{4} - 3959612 T^{5} + 55142006 T^{6} - 659168568 T^{7} + 7454929856 T^{8} - 74328091409 T^{9} + 701127963032 T^{10} - 74328091409 p T^{11} + 7454929856 p^{2} T^{12} - 659168568 p^{3} T^{13} + 55142006 p^{4} T^{14} - 3959612 p^{5} T^{15} + 270063 p^{6} T^{16} - 14711 p^{7} T^{17} + 782 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 14 T + 577 T^{2} + 5800 T^{3} + 138728 T^{4} + 1019691 T^{5} + 19211234 T^{6} + 101335232 T^{7} + 1848101498 T^{8} + 7313651899 T^{9} + 153849005048 T^{10} + 7313651899 p T^{11} + 1848101498 p^{2} T^{12} + 101335232 p^{3} T^{13} + 19211234 p^{4} T^{14} + 1019691 p^{5} T^{15} + 138728 p^{6} T^{16} + 5800 p^{7} T^{17} + 577 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 31 T + 1084 T^{2} + 21819 T^{3} + 452262 T^{4} + 6916306 T^{5} + 106735781 T^{6} + 1322186569 T^{7} + 16410110830 T^{8} + 169159033935 T^{9} + 1742700155052 T^{10} + 169159033935 p T^{11} + 16410110830 p^{2} T^{12} + 1322186569 p^{3} T^{13} + 106735781 p^{4} T^{14} + 6916306 p^{5} T^{15} + 452262 p^{6} T^{16} + 21819 p^{7} T^{17} + 1084 p^{8} T^{18} + 31 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 32 T + 740 T^{2} + 12795 T^{3} + 186194 T^{4} + 2199458 T^{5} + 22630839 T^{6} + 194197764 T^{7} + 1475234704 T^{8} + 10082483539 T^{9} + 85871535452 T^{10} + 10082483539 p T^{11} + 1475234704 p^{2} T^{12} + 194197764 p^{3} T^{13} + 22630839 p^{4} T^{14} + 2199458 p^{5} T^{15} + 186194 p^{6} T^{16} + 12795 p^{7} T^{17} + 740 p^{8} T^{18} + 32 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.49472255342800592748067931807, −3.44482222992153179654229613831, −3.41000186456630771158030012682, −3.33695640703528301999421908205, −3.24780639095206052665322855110, −2.80469873886323260491954814055, −2.77027600087297631741941913532, −2.55841507797118741975635694092, −2.53469550356192433916511095542, −2.50693743221042700471138405652, −2.42554632453823099519741862492, −2.40527645155454940802793754072, −2.24210525531034922227719843401, −2.23570714394053259578544156867, −2.17033578062647710872893260911, −1.53344292590671643553213393521, −1.48999115758798509936581458028, −1.47976269308042810957422518239, −1.45104423453994539580849783908, −1.36578187953052818518468729009, −1.21181521991409156129616521081, −1.17828345971793643390830037949, −1.10888687384730257824617762195, −1.07960617300720810063433122728, −0.945475395247190560206654767374, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.945475395247190560206654767374, 1.07960617300720810063433122728, 1.10888687384730257824617762195, 1.17828345971793643390830037949, 1.21181521991409156129616521081, 1.36578187953052818518468729009, 1.45104423453994539580849783908, 1.47976269308042810957422518239, 1.48999115758798509936581458028, 1.53344292590671643553213393521, 2.17033578062647710872893260911, 2.23570714394053259578544156867, 2.24210525531034922227719843401, 2.40527645155454940802793754072, 2.42554632453823099519741862492, 2.50693743221042700471138405652, 2.53469550356192433916511095542, 2.55841507797118741975635694092, 2.77027600087297631741941913532, 2.80469873886323260491954814055, 3.24780639095206052665322855110, 3.33695640703528301999421908205, 3.41000186456630771158030012682, 3.44482222992153179654229613831, 3.49472255342800592748067931807
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.