Dirichlet series
| L(s) = 1 | − 5·2-s + 3-s + 10·4-s + 5·5-s − 5·6-s + 5·7-s − 5·8-s + 9-s − 25·10-s + 3·11-s + 10·12-s + 8·13-s − 25·14-s + 5·15-s − 20·16-s − 2·17-s − 5·18-s − 2·19-s + 50·20-s + 5·21-s − 15·22-s − 5·23-s − 5·24-s + 25·25-s − 40·26-s − 27-s + 50·28-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 0.577·3-s + 5·4-s + 2.23·5-s − 2.04·6-s + 1.88·7-s − 1.76·8-s + 1/3·9-s − 7.90·10-s + 0.904·11-s + 2.88·12-s + 2.21·13-s − 6.68·14-s + 1.29·15-s − 5·16-s − 0.485·17-s − 1.17·18-s − 0.458·19-s + 11.1·20-s + 1.09·21-s − 3.19·22-s − 1.04·23-s − 1.02·24-s + 5·25-s − 7.84·26-s − 0.192·27-s + 9.44·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{20} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(155874.\) |
| Root analytic conductor: | \(1.81818\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.850511098\) |
| \(L(\frac12)\) | \(\approx\) | \(1.850511098\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{5} \) |
| 3 | \( 1 - T + 2 T^{3} - 2 T^{4} - p T^{5} - 2 p T^{6} + 2 p^{2} T^{7} - p^{4} T^{9} + p^{5} T^{10} \) | |
| 23 | \( ( 1 + T + T^{2} )^{5} \) | |
| good | 5 | \( 1 - p T + 29 T^{3} + 11 T^{4} - 6 p^{2} T^{5} - 81 T^{6} + 467 T^{7} + 286 T^{8} + 179 p T^{9} - 1541 p T^{10} + 179 p^{2} T^{11} + 286 p^{2} T^{12} + 467 p^{3} T^{13} - 81 p^{4} T^{14} - 6 p^{7} T^{15} + 11 p^{6} T^{16} + 29 p^{7} T^{17} - p^{10} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 5 T - 3 T^{2} + 16 T^{3} + 128 T^{4} - 107 T^{5} - 555 T^{6} - 176 p T^{7} + 1762 T^{8} + 624 p T^{9} + 7473 T^{10} + 624 p^{2} T^{11} + 1762 p^{2} T^{12} - 176 p^{4} T^{13} - 555 p^{4} T^{14} - 107 p^{5} T^{15} + 128 p^{6} T^{16} + 16 p^{7} T^{17} - 3 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 3 T - 9 T^{2} - 30 T^{3} + 40 T^{4} + 81 p T^{5} - 7 T^{6} + 1104 T^{7} - 7418 T^{8} - 39180 T^{9} + 20861 T^{10} - 39180 p T^{11} - 7418 p^{2} T^{12} + 1104 p^{3} T^{13} - 7 p^{4} T^{14} + 81 p^{6} T^{15} + 40 p^{6} T^{16} - 30 p^{7} T^{17} - 9 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 8 T + 3 T^{2} + 190 T^{3} - 607 T^{4} - 914 T^{5} + 57 p^{2} T^{6} - 30077 T^{7} + 47758 T^{8} + 290457 T^{9} - 2119419 T^{10} + 290457 p T^{11} + 47758 p^{2} T^{12} - 30077 p^{3} T^{13} + 57 p^{6} T^{14} - 914 p^{5} T^{15} - 607 p^{6} T^{16} + 190 p^{7} T^{17} + 3 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( ( 1 + T + 40 T^{2} + 54 T^{3} + 1052 T^{4} + 757 T^{5} + 1052 p T^{6} + 54 p^{2} T^{7} + 40 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 19 | \( ( 1 + T + 2 T^{2} - 37 T^{3} + 17 T^{4} - 85 T^{5} + 17 p T^{6} - 37 p^{2} T^{7} + 2 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 29 | \( 1 - 18 T + 96 T^{2} - 126 T^{3} + 1648 T^{4} - 14673 T^{5} - 33949 T^{6} + 321057 T^{7} + 1915075 T^{8} - 6088074 T^{9} - 42765901 T^{10} - 6088074 p T^{11} + 1915075 p^{2} T^{12} + 321057 p^{3} T^{13} - 33949 p^{4} T^{14} - 14673 p^{5} T^{15} + 1648 p^{6} T^{16} - 126 p^{7} T^{17} + 96 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 8 T + 35 T^{2} + 182 T^{3} - 2415 T^{4} + 11530 T^{5} + 31113 T^{6} - 459041 T^{7} + 1770624 T^{8} + 9723141 T^{9} - 93087433 T^{10} + 9723141 p T^{11} + 1770624 p^{2} T^{12} - 459041 p^{3} T^{13} + 31113 p^{4} T^{14} + 11530 p^{5} T^{15} - 2415 p^{6} T^{16} + 182 p^{7} T^{17} + 35 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( ( 1 + 6 T + 130 T^{2} + 633 T^{3} + 8420 T^{4} + 32457 T^{5} + 8420 p T^{6} + 633 p^{2} T^{7} + 130 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 - 24 T + 273 T^{2} - 1862 T^{3} + 7217 T^{4} - 3930 T^{5} - 116191 T^{6} + 796913 T^{7} - 7804250 T^{8} + 90867809 T^{9} - 724266287 T^{10} + 90867809 p T^{11} - 7804250 p^{2} T^{12} + 796913 p^{3} T^{13} - 116191 p^{4} T^{14} - 3930 p^{5} T^{15} + 7217 p^{6} T^{16} - 1862 p^{7} T^{17} + 273 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 11 T - 36 T^{2} - 979 T^{3} - 2173 T^{4} + 26472 T^{5} + 139141 T^{6} + 215685 T^{7} + 706990 T^{8} - 18508015 T^{9} - 270880641 T^{10} - 18508015 p T^{11} + 706990 p^{2} T^{12} + 215685 p^{3} T^{13} + 139141 p^{4} T^{14} + 26472 p^{5} T^{15} - 2173 p^{6} T^{16} - 979 p^{7} T^{17} - 36 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 9 T - 144 T^{2} + 1003 T^{3} + 16406 T^{4} - 73083 T^{5} - 1333450 T^{6} + 3165083 T^{7} + 87036169 T^{8} - 67472230 T^{9} - 4476295436 T^{10} - 67472230 p T^{11} + 87036169 p^{2} T^{12} + 3165083 p^{3} T^{13} - 1333450 p^{4} T^{14} - 73083 p^{5} T^{15} + 16406 p^{6} T^{16} + 1003 p^{7} T^{17} - 144 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( ( 1 + 29 T + 588 T^{2} + 7865 T^{3} + 83747 T^{4} + 674715 T^{5} + 83747 p T^{6} + 7865 p^{2} T^{7} + 588 p^{3} T^{8} + 29 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 59 | \( 1 - 21 T + 195 T^{2} - 636 T^{3} - 11138 T^{4} + 241863 T^{5} - 2419552 T^{6} + 12348699 T^{7} + 29544946 T^{8} - 1176739077 T^{9} + 11938831289 T^{10} - 1176739077 p T^{11} + 29544946 p^{2} T^{12} + 12348699 p^{3} T^{13} - 2419552 p^{4} T^{14} + 241863 p^{5} T^{15} - 11138 p^{6} T^{16} - 636 p^{7} T^{17} + 195 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 17 T + 45 T^{2} - 248 T^{3} + 7988 T^{4} + 26035 T^{5} - 544905 T^{6} - 1846640 T^{7} + 14760358 T^{8} - 140103600 T^{9} + 2685211947 T^{10} - 140103600 p T^{11} + 14760358 p^{2} T^{12} - 1846640 p^{3} T^{13} - 544905 p^{4} T^{14} + 26035 p^{5} T^{15} + 7988 p^{6} T^{16} - 248 p^{7} T^{17} + 45 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 3 T - 131 T^{2} + 1332 T^{3} + 2574 T^{4} - 106857 T^{5} + 469173 T^{6} + 330390 T^{7} - 8113452 T^{8} + 117579348 T^{9} - 2090521503 T^{10} + 117579348 p T^{11} - 8113452 p^{2} T^{12} + 330390 p^{3} T^{13} + 469173 p^{4} T^{14} - 106857 p^{5} T^{15} + 2574 p^{6} T^{16} + 1332 p^{7} T^{17} - 131 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 + 9 T + 153 T^{2} + 681 T^{3} + 9905 T^{4} + 23043 T^{5} + 9905 p T^{6} + 681 p^{2} T^{7} + 153 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( ( 1 + 7 T + 268 T^{2} + 1275 T^{3} + 31079 T^{4} + 111455 T^{5} + 31079 p T^{6} + 1275 p^{2} T^{7} + 268 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( 1 - 15 T - 250 T^{2} + 2721 T^{3} + 70247 T^{4} - 483072 T^{5} - 10469587 T^{6} + 32377143 T^{7} + 1367938078 T^{8} - 1675777869 T^{9} - 116924120025 T^{10} - 1675777869 p T^{11} + 1367938078 p^{2} T^{12} + 32377143 p^{3} T^{13} - 10469587 p^{4} T^{14} - 483072 p^{5} T^{15} + 70247 p^{6} T^{16} + 2721 p^{7} T^{17} - 250 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 21 T - 92 T^{2} + 2709 T^{3} + 41107 T^{4} - 493448 T^{5} - 5410203 T^{6} + 28687709 T^{7} + 826989782 T^{8} - 1715119167 T^{9} - 69151581921 T^{10} - 1715119167 p T^{11} + 826989782 p^{2} T^{12} + 28687709 p^{3} T^{13} - 5410203 p^{4} T^{14} - 493448 p^{5} T^{15} + 41107 p^{6} T^{16} + 2709 p^{7} T^{17} - 92 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( ( 1 + 9 T + 330 T^{2} + 2931 T^{3} + 49727 T^{4} + 378291 T^{5} + 49727 p T^{6} + 2931 p^{2} T^{7} + 330 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 97 | \( 1 + 32 T + 300 T^{2} + 1082 T^{3} + 26204 T^{4} + 248079 T^{5} - 2914943 T^{6} - 32764569 T^{7} + 99563731 T^{8} - 1149622528 T^{9} - 46599100017 T^{10} - 1149622528 p T^{11} + 99563731 p^{2} T^{12} - 32764569 p^{3} T^{13} - 2914943 p^{4} T^{14} + 248079 p^{5} T^{15} + 26204 p^{6} T^{16} + 1082 p^{7} T^{17} + 300 p^{8} T^{18} + 32 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.35022793925545223922576179711, −4.33874576746438107105843356221, −4.10345573663123232588973616543, −3.87095486367903148228756230112, −3.66591092797131336060302252929, −3.61518540696669415574876752478, −3.60274869710491767694107889271, −3.32729291849535757563414323756, −3.17267279565813373502820158787, −2.85264013709632686222730980805, −2.81996177160334691251047378060, −2.77757909727126290327485797449, −2.56307929554595840624920767424, −2.38952966075333760496150237564, −2.28571832839473440178037363983, −2.06673177915636443911724598170, −1.86916581949056982266069648125, −1.86138002185800806279822413499, −1.46438208488221911242876761172, −1.30610656524558492923941960777, −1.27229357327223662075674808017, −1.18875869824407759720920172576, −1.11409354222663885549296535289, −0.849478866672950339283296662898, −0.36717894944553653146920656347, 0.36717894944553653146920656347, 0.849478866672950339283296662898, 1.11409354222663885549296535289, 1.18875869824407759720920172576, 1.27229357327223662075674808017, 1.30610656524558492923941960777, 1.46438208488221911242876761172, 1.86138002185800806279822413499, 1.86916581949056982266069648125, 2.06673177915636443911724598170, 2.28571832839473440178037363983, 2.38952966075333760496150237564, 2.56307929554595840624920767424, 2.77757909727126290327485797449, 2.81996177160334691251047378060, 2.85264013709632686222730980805, 3.17267279565813373502820158787, 3.32729291849535757563414323756, 3.60274869710491767694107889271, 3.61518540696669415574876752478, 3.66591092797131336060302252929, 3.87095486367903148228756230112, 4.10345573663123232588973616543, 4.33874576746438107105843356221, 4.35022793925545223922576179711
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.