Dirichlet series
| L(s) = 1 | + 4·2-s + 3-s + 2·4-s − 3·5-s + 4·6-s + 6·7-s − 11·8-s − 12·10-s − 15·11-s + 2·12-s + 8·13-s + 24·14-s − 3·15-s − 11·16-s + 17-s − 9·19-s − 6·20-s + 6·21-s − 60·22-s + 21·23-s − 11·24-s + 5·25-s + 32·26-s + 12·28-s − 8·29-s − 12·30-s − 23·31-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.577·3-s + 4-s − 1.34·5-s + 1.63·6-s + 2.26·7-s − 3.88·8-s − 3.79·10-s − 4.52·11-s + 0.577·12-s + 2.21·13-s + 6.41·14-s − 0.774·15-s − 2.75·16-s + 0.242·17-s − 2.06·19-s − 1.34·20-s + 1.30·21-s − 12.7·22-s + 4.37·23-s − 2.24·24-s + 25-s + 6.27·26-s + 2.26·28-s − 1.48·29-s − 2.19·30-s − 4.13·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00257787\) |
| Root analytic conductor: | \(0.742272\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.8803536029\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8803536029\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 23 | \( 1 - 21 T + 221 T^{2} - 1627 T^{3} + 9505 T^{4} - 47937 T^{5} + 9505 p T^{6} - 1627 p^{2} T^{7} + 221 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 - p^{2} T + 7 p T^{2} - 37 T^{3} + 87 T^{4} - 175 T^{5} + 41 p^{3} T^{6} - 555 T^{7} + 893 T^{8} - 335 p^{2} T^{9} + 1957 T^{10} - 335 p^{3} T^{11} + 893 p^{2} T^{12} - 555 p^{3} T^{13} + 41 p^{7} T^{14} - 175 p^{5} T^{15} + 87 p^{6} T^{16} - 37 p^{7} T^{17} + 7 p^{9} T^{18} - p^{11} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + 3 T + 4 T^{2} - 14 T^{3} - 8 p T^{4} - 28 T^{5} + 204 T^{6} + 466 T^{7} + 367 T^{8} - 987 T^{9} - 3344 T^{10} - 987 p T^{11} + 367 p^{2} T^{12} + 466 p^{3} T^{13} + 204 p^{4} T^{14} - 28 p^{5} T^{15} - 8 p^{7} T^{16} - 14 p^{7} T^{17} + 4 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - 6 T + 29 T^{2} - 88 T^{3} + 303 T^{4} - 850 T^{5} + 3089 T^{6} - 8580 T^{7} + 26315 T^{8} - 1240 p^{2} T^{9} + 176681 T^{10} - 1240 p^{3} T^{11} + 26315 p^{2} T^{12} - 8580 p^{3} T^{13} + 3089 p^{4} T^{14} - 850 p^{5} T^{15} + 303 p^{6} T^{16} - 88 p^{7} T^{17} + 29 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 15 T + 126 T^{2} + 779 T^{3} + 3721 T^{4} + 13586 T^{5} + 34027 T^{6} + 24755 T^{7} - 299730 T^{8} - 2077875 T^{9} - 8295319 T^{10} - 2077875 p T^{11} - 299730 p^{2} T^{12} + 24755 p^{3} T^{13} + 34027 p^{4} T^{14} + 13586 p^{5} T^{15} + 3721 p^{6} T^{16} + 779 p^{7} T^{17} + 126 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 8 T + 18 T^{2} + 2 p T^{3} - 343 T^{4} + 1427 T^{5} - 3437 T^{6} + 2631 T^{7} + 32477 T^{8} - 137588 T^{9} + 302017 T^{10} - 137588 p T^{11} + 32477 p^{2} T^{12} + 2631 p^{3} T^{13} - 3437 p^{4} T^{14} + 1427 p^{5} T^{15} - 343 p^{6} T^{16} + 2 p^{8} T^{17} + 18 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - T - 16 T^{2} + 110 T^{3} + 162 T^{4} - 2989 T^{5} + 6032 T^{6} + 2629 p T^{7} - 232113 T^{8} - 394161 T^{9} + 4546981 T^{10} - 394161 p T^{11} - 232113 p^{2} T^{12} + 2629 p^{4} T^{13} + 6032 p^{4} T^{14} - 2989 p^{5} T^{15} + 162 p^{6} T^{16} + 110 p^{7} T^{17} - 16 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 9 T + 29 T^{2} - 9 T^{3} - 104 T^{4} + 3151 T^{5} + 22250 T^{6} + 47211 T^{7} + 63925 T^{8} + 1232132 T^{9} + 9187685 T^{10} + 1232132 p T^{11} + 63925 p^{2} T^{12} + 47211 p^{3} T^{13} + 22250 p^{4} T^{14} + 3151 p^{5} T^{15} - 104 p^{6} T^{16} - 9 p^{7} T^{17} + 29 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 8 T + 79 T^{2} + 279 T^{3} + 3054 T^{4} + 8311 T^{5} + 77758 T^{6} - 142159 T^{7} + 552841 T^{8} - 11551859 T^{9} + 4707627 T^{10} - 11551859 p T^{11} + 552841 p^{2} T^{12} - 142159 p^{3} T^{13} + 77758 p^{4} T^{14} + 8311 p^{5} T^{15} + 3054 p^{6} T^{16} + 279 p^{7} T^{17} + 79 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 23 T + 267 T^{2} + 2425 T^{3} + 19316 T^{4} + 132549 T^{5} + 838837 T^{6} + 5077982 T^{7} + 28910514 T^{8} + 164808391 T^{9} + 941710661 T^{10} + 164808391 p T^{11} + 28910514 p^{2} T^{12} + 5077982 p^{3} T^{13} + 838837 p^{4} T^{14} + 132549 p^{5} T^{15} + 19316 p^{6} T^{16} + 2425 p^{7} T^{17} + 267 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 3 T - 94 T^{2} + 492 T^{3} + 4103 T^{4} - 27059 T^{5} - 94086 T^{6} + 750810 T^{7} + 1866026 T^{8} - 8908504 T^{9} - 61696579 T^{10} - 8908504 p T^{11} + 1866026 p^{2} T^{12} + 750810 p^{3} T^{13} - 94086 p^{4} T^{14} - 27059 p^{5} T^{15} + 4103 p^{6} T^{16} + 492 p^{7} T^{17} - 94 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 15 T + 96 T^{2} + 495 T^{3} + 805 T^{4} - 11542 T^{5} - 34425 T^{6} - 120923 T^{7} - 2976552 T^{8} - 36113963 T^{9} - 311834645 T^{10} - 36113963 p T^{11} - 2976552 p^{2} T^{12} - 120923 p^{3} T^{13} - 34425 p^{4} T^{14} - 11542 p^{5} T^{15} + 805 p^{6} T^{16} + 495 p^{7} T^{17} + 96 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 22 T + 177 T^{2} + 187 T^{3} - 16312 T^{4} + 138105 T^{5} - 236708 T^{6} - 101299 p T^{7} + 32555403 T^{8} - 6641921 T^{9} - 819427927 T^{10} - 6641921 p T^{11} + 32555403 p^{2} T^{12} - 101299 p^{4} T^{13} - 236708 p^{4} T^{14} + 138105 p^{5} T^{15} - 16312 p^{6} T^{16} + 187 p^{7} T^{17} + 177 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 - 2 T + 87 T^{2} - 11 T^{3} + 6286 T^{4} - 5331 T^{5} + 6286 p T^{6} - 11 p^{2} T^{7} + 87 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 - 29 T + 381 T^{2} - 3143 T^{3} + 18176 T^{4} - 77121 T^{5} + 345100 T^{6} - 3675145 T^{7} + 50283377 T^{8} - 508562448 T^{9} + 4010222963 T^{10} - 508562448 p T^{11} + 50283377 p^{2} T^{12} - 3675145 p^{3} T^{13} + 345100 p^{4} T^{14} - 77121 p^{5} T^{15} + 18176 p^{6} T^{16} - 3143 p^{7} T^{17} + 381 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 54 T + 1372 T^{2} + 22018 T^{3} + 252070 T^{4} + 2136486 T^{5} + 12031747 T^{6} + 8335314 T^{7} - 798790900 T^{8} - 11779154714 T^{9} - 107354265811 T^{10} - 11779154714 p T^{11} - 798790900 p^{2} T^{12} + 8335314 p^{3} T^{13} + 12031747 p^{4} T^{14} + 2136486 p^{5} T^{15} + 252070 p^{6} T^{16} + 22018 p^{7} T^{17} + 1372 p^{8} T^{18} + 54 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 30 T + 443 T^{2} + 4464 T^{3} + 37531 T^{4} + 272683 T^{5} + 1275059 T^{6} - 2684292 T^{7} - 124836911 T^{8} - 1476855385 T^{9} - 12668990665 T^{10} - 1476855385 p T^{11} - 124836911 p^{2} T^{12} - 2684292 p^{3} T^{13} + 1275059 p^{4} T^{14} + 272683 p^{5} T^{15} + 37531 p^{6} T^{16} + 4464 p^{7} T^{17} + 443 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - T - 99 T^{2} + 265 T^{3} + 4608 T^{4} + 5049 T^{5} - 167111 T^{6} - 1321046 T^{7} + 818642 T^{8} + 31175519 T^{9} + 660099725 T^{10} + 31175519 p T^{11} + 818642 p^{2} T^{12} - 1321046 p^{3} T^{13} - 167111 p^{4} T^{14} + 5049 p^{5} T^{15} + 4608 p^{6} T^{16} + 265 p^{7} T^{17} - 99 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 3 T + 92 T^{2} - 586 T^{3} + 4976 T^{4} - 56007 T^{5} + 816294 T^{6} - 2120279 T^{7} + 77478515 T^{8} - 410651139 T^{9} + 3683983919 T^{10} - 410651139 p T^{11} + 77478515 p^{2} T^{12} - 2120279 p^{3} T^{13} + 816294 p^{4} T^{14} - 56007 p^{5} T^{15} + 4976 p^{6} T^{16} - 586 p^{7} T^{17} + 92 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 47 T + 1124 T^{2} + 18454 T^{3} + 229148 T^{4} + 2160317 T^{5} + 13916836 T^{6} + 24276170 T^{7} - 850705864 T^{8} - 15034376302 T^{9} - 154794035109 T^{10} - 15034376302 p T^{11} - 850705864 p^{2} T^{12} + 24276170 p^{3} T^{13} + 13916836 p^{4} T^{14} + 2160317 p^{5} T^{15} + 229148 p^{6} T^{16} + 18454 p^{7} T^{17} + 1124 p^{8} T^{18} + 47 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 18 T - 63 T^{2} + 4668 T^{3} - 31978 T^{4} - 362792 T^{5} + 5690496 T^{6} + 694138 T^{7} - 383558844 T^{8} + 618029740 T^{9} + 20150719357 T^{10} + 618029740 p T^{11} - 383558844 p^{2} T^{12} + 694138 p^{3} T^{13} + 5690496 p^{4} T^{14} - 362792 p^{5} T^{15} - 31978 p^{6} T^{16} + 4668 p^{7} T^{17} - 63 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 18 T + 252 T^{2} - 3460 T^{3} + 32036 T^{4} - 300798 T^{5} + 3210347 T^{6} - 29964434 T^{7} + 364683526 T^{8} - 3965360938 T^{9} + 35460911113 T^{10} - 3965360938 p T^{11} + 364683526 p^{2} T^{12} - 29964434 p^{3} T^{13} + 3210347 p^{4} T^{14} - 300798 p^{5} T^{15} + 32036 p^{6} T^{16} - 3460 p^{7} T^{17} + 252 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 25 T + 349 T^{2} - 5433 T^{3} + 70257 T^{4} - 636571 T^{5} + 5023208 T^{6} - 31863191 T^{7} + 72717556 T^{8} + 1091439866 T^{9} - 16621970441 T^{10} + 1091439866 p T^{11} + 72717556 p^{2} T^{12} - 31863191 p^{3} T^{13} + 5023208 p^{4} T^{14} - 636571 p^{5} T^{15} + 70257 p^{6} T^{16} - 5433 p^{7} T^{17} + 349 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 21 T + 113 T^{2} + 500 T^{3} + 7337 T^{4} - 408365 T^{5} + 3956908 T^{6} - 8362071 T^{7} + 11080284 T^{8} - 2949238215 T^{9} + 50457844339 T^{10} - 2949238215 p T^{11} + 11080284 p^{2} T^{12} - 8362071 p^{3} T^{13} + 3956908 p^{4} T^{14} - 408365 p^{5} T^{15} + 7337 p^{6} T^{16} + 500 p^{7} T^{17} + 113 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−5.83762155503883520006997099509, −5.83506275083670387637702373807, −5.68749855274324005236971184280, −5.58770271701022535878679679935, −5.55893831959125006122150709896, −5.46312092844551907230211753743, −4.97372659023992932617676020665, −4.90049856169907381687740101634, −4.88740516858702272645932748937, −4.79844477401519782716514577998, −4.62869899383640465896629418246, −4.61629221355278721525381380927, −4.50148315843682922811825352415, −4.18266752064744035927116549833, −4.03423105257628206892948813824, −3.94555675701795956501059105764, −3.57112208244166780267811491598, −3.37282705052312990836249607613, −3.27533312150018725849889219813, −3.17786159089329656973945215190, −2.85783205743221049117445864961, −2.79570899128405437725245892643, −2.06804246159223874210557656941, −1.76343600730618198922411243118, −1.75989173279955733901018825534, 1.75989173279955733901018825534, 1.76343600730618198922411243118, 2.06804246159223874210557656941, 2.79570899128405437725245892643, 2.85783205743221049117445864961, 3.17786159089329656973945215190, 3.27533312150018725849889219813, 3.37282705052312990836249607613, 3.57112208244166780267811491598, 3.94555675701795956501059105764, 4.03423105257628206892948813824, 4.18266752064744035927116549833, 4.50148315843682922811825352415, 4.61629221355278721525381380927, 4.62869899383640465896629418246, 4.79844477401519782716514577998, 4.88740516858702272645932748937, 4.90049856169907381687740101634, 4.97372659023992932617676020665, 5.46312092844551907230211753743, 5.55893831959125006122150709896, 5.58770271701022535878679679935, 5.68749855274324005236971184280, 5.83506275083670387637702373807, 5.83762155503883520006997099509
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.