Dirichlet series
| L(s) = 1 | + 3-s + 4·5-s + 4·7-s + 6·9-s − 4·11-s − 8·13-s + 4·15-s + 12·17-s − 19-s + 4·21-s − 3·23-s + 20·25-s + 8·27-s + 7·29-s − 6·31-s − 4·33-s + 16·35-s − 8·39-s + 5·41-s + 7·43-s + 24·45-s + 54·47-s + 4·49-s + 12·51-s − 21·53-s − 16·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1.51·7-s + 2·9-s − 1.20·11-s − 2.21·13-s + 1.03·15-s + 2.91·17-s − 0.229·19-s + 0.872·21-s − 0.625·23-s + 4·25-s + 1.53·27-s + 1.29·29-s − 1.07·31-s − 0.696·33-s + 2.70·35-s − 1.28·39-s + 0.780·41-s + 1.06·43-s + 3.57·45-s + 7.87·47-s + 4/7·49-s + 1.68·51-s − 2.88·53-s − 2.15·55-s − 0.132·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{40} \cdot 3^{20} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.14123\times 10^{9}\) |
| Root analytic conductor: | \(2.83706\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{40} \cdot 3^{20} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(66.82157909\) |
| \(L(\frac12)\) | \(\approx\) | \(66.82157909\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - T - 5 T^{2} + p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} + p^{3} T^{7} - 5 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | |
| 7 | \( 1 - 4 T + 12 T^{2} - 47 T^{3} + 146 T^{4} - 309 T^{5} + 146 p T^{6} - 47 p^{2} T^{7} + 12 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 5 | \( 1 - 4 T - 4 T^{2} + 44 T^{3} - 41 T^{4} - 119 T^{5} + 222 T^{6} - 456 T^{7} + 1623 T^{8} + 2021 T^{9} - 16541 T^{10} + 2021 p T^{11} + 1623 p^{2} T^{12} - 456 p^{3} T^{13} + 222 p^{4} T^{14} - 119 p^{5} T^{15} - 41 p^{6} T^{16} + 44 p^{7} T^{17} - 4 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) |
| 11 | \( 1 + 4 T - 31 T^{2} - 134 T^{3} + 607 T^{4} + 2492 T^{5} - 8385 T^{6} - 27495 T^{7} + 98940 T^{8} + 135733 T^{9} - 1043873 T^{10} + 135733 p T^{11} + 98940 p^{2} T^{12} - 27495 p^{3} T^{13} - 8385 p^{4} T^{14} + 2492 p^{5} T^{15} + 607 p^{6} T^{16} - 134 p^{7} T^{17} - 31 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 8 T - 14 T^{2} - 14 p T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 3323 p T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 358672 p T^{11} + 305249 p^{2} T^{12} - 3323 p^{4} T^{13} - 12871 p^{4} T^{14} + 4429 p^{5} T^{15} + 686 p^{6} T^{16} - 14 p^{8} T^{17} - 14 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 2436261 p T^{11} + 1002861 p^{2} T^{12} + 65736 p^{3} T^{13} - 6180 p^{4} T^{14} - 11847 p^{5} T^{15} + 1185 p^{6} T^{16} + 192 p^{7} T^{17} + 14 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + T - 53 T^{2} - 10 p T^{3} + 1262 T^{4} + 7007 T^{5} - 13111 T^{6} - 116110 T^{7} + 67964 T^{8} + 721616 T^{9} - 440023 T^{10} + 721616 p T^{11} + 67964 p^{2} T^{12} - 116110 p^{3} T^{13} - 13111 p^{4} T^{14} + 7007 p^{5} T^{15} + 1262 p^{6} T^{16} - 10 p^{8} T^{17} - 53 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 3 T - 43 T^{2} - 294 T^{3} + 6 T^{4} + 5127 T^{5} + 21792 T^{6} + 135027 T^{7} + 502362 T^{8} - 3271749 T^{9} - 33095343 T^{10} - 3271749 p T^{11} + 502362 p^{2} T^{12} + 135027 p^{3} T^{13} + 21792 p^{4} T^{14} + 5127 p^{5} T^{15} + 6 p^{6} T^{16} - 294 p^{7} T^{17} - 43 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 2570041 p T^{11} + 6918636 p^{2} T^{12} + 341373 p^{3} T^{13} - 199563 p^{4} T^{14} - 15146 p^{5} T^{15} + 4561 p^{6} T^{16} + 419 p^{7} T^{17} - 76 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( ( 1 + 3 T + 134 T^{2} + 308 T^{3} + 250 p T^{4} + 13615 T^{5} + 250 p^{2} T^{6} + 308 p^{2} T^{7} + 134 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 37 | \( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 33131096 p T^{11} - 9801827 p^{2} T^{12} + 2296536 p^{3} T^{13} + 27130 p^{4} T^{14} - 45352 p^{5} T^{15} + 4503 p^{6} T^{16} + 560 p^{7} T^{17} - 89 p^{8} T^{18} + p^{10} T^{20} \) | |
| 41 | \( 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 41392439 p T^{11} + 23706084 p^{2} T^{12} + 2345451 p^{3} T^{13} - 554055 p^{4} T^{14} - 54412 p^{5} T^{15} + 10507 p^{6} T^{16} + 733 p^{7} T^{17} - 136 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 7 T - 77 T^{2} + 66 T^{3} + 7014 T^{4} + 3843 T^{5} - 95427 T^{6} - 1632678 T^{7} - 3708600 T^{8} + 15416324 T^{9} + 670279801 T^{10} + 15416324 p T^{11} - 3708600 p^{2} T^{12} - 1632678 p^{3} T^{13} - 95427 p^{4} T^{14} + 3843 p^{5} T^{15} + 7014 p^{6} T^{16} + 66 p^{7} T^{17} - 77 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 - 27 T + 448 T^{2} - 5169 T^{3} + 48091 T^{4} - 359985 T^{5} + 48091 p T^{6} - 5169 p^{2} T^{7} + 448 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 340111866 p T^{11} + 110973258 p^{2} T^{12} - 3783942 p^{3} T^{13} - 601755 p^{4} T^{14} + 177027 p^{5} T^{15} + 12966 p^{6} T^{16} - 924 p^{7} T^{17} + 41 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( ( 1 - 30 T + 601 T^{2} - 8193 T^{3} + 88864 T^{4} - 752289 T^{5} + 88864 p T^{6} - 8193 p^{2} T^{7} + 601 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 61 | \( ( 1 - 14 T + 339 T^{2} - 3409 T^{3} + 43418 T^{4} - 311709 T^{5} + 43418 p T^{6} - 3409 p^{2} T^{7} + 339 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 67 | \( ( 1 + 2 T + 132 T^{2} + 196 T^{3} + 10871 T^{4} + 15429 T^{5} + 10871 p T^{6} + 196 p^{2} T^{7} + 132 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 71 | \( ( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 15679 p T^{6} - 285 p^{2} T^{7} + 187 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 238799411 p T^{11} + 288508378 p^{2} T^{12} + 9021201 p^{3} T^{13} - 2002535 p^{4} T^{14} - 235276 p^{5} T^{15} + 16563 p^{6} T^{16} + 2501 p^{7} T^{17} - 134 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( ( 1 + 4 T + 300 T^{2} + 1488 T^{3} + 39873 T^{4} + 184983 T^{5} + 39873 p T^{6} + 1488 p^{2} T^{7} + 300 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 83 | \( 1 + 9 T - 148 T^{2} + 297 T^{3} + 24654 T^{4} - 118125 T^{5} - 807174 T^{6} + 21382137 T^{7} - 37648479 T^{8} - 452536146 T^{9} + 15509586612 T^{10} - 452536146 p T^{11} - 37648479 p^{2} T^{12} + 21382137 p^{3} T^{13} - 807174 p^{4} T^{14} - 118125 p^{5} T^{15} + 24654 p^{6} T^{16} + 297 p^{7} T^{17} - 148 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 3271180573 p T^{11} + 740914101 p^{2} T^{12} + 18973932 p^{3} T^{13} - 1780638 p^{4} T^{14} - 611939 p^{5} T^{15} + 31273 p^{6} T^{16} + 1736 p^{7} T^{17} + 104 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 908843245 p T^{11} + 472663750 p^{2} T^{12} - 6881349 p^{3} T^{13} - 4545035 p^{4} T^{14} + 14090 p^{5} T^{15} + 27813 p^{6} T^{16} - 1534 p^{7} T^{17} - 197 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.70029420914446370702396553005, −3.69642135781798168197581543462, −3.66843940760103763017763795582, −3.35611019471545829189951929903, −3.23698888432409870110642165395, −2.80402480315918662210015589253, −2.72010998386897578700393255373, −2.65998822119982972169648103332, −2.60880612043996317595819748511, −2.52240966077642653411811289599, −2.43227705628610174409541359121, −2.39756355375532837165302699089, −2.31121860448573915087238130365, −2.30050884823586810539645575230, −2.20793891611074692591683017340, −1.75448173981580667955203951564, −1.72783622890494482173941500225, −1.48848888189180257337448790869, −1.34545836786369813739461595908, −1.14315807826902459644934058784, −1.06580315227596789057935993124, −1.03163659746796051064322069071, −0.75938539082710144559210803111, −0.67432416799957010167286783723, −0.45606050691765940457999696647, 0.45606050691765940457999696647, 0.67432416799957010167286783723, 0.75938539082710144559210803111, 1.03163659746796051064322069071, 1.06580315227596789057935993124, 1.14315807826902459644934058784, 1.34545836786369813739461595908, 1.48848888189180257337448790869, 1.72783622890494482173941500225, 1.75448173981580667955203951564, 2.20793891611074692591683017340, 2.30050884823586810539645575230, 2.31121860448573915087238130365, 2.39756355375532837165302699089, 2.43227705628610174409541359121, 2.52240966077642653411811289599, 2.60880612043996317595819748511, 2.65998822119982972169648103332, 2.72010998386897578700393255373, 2.80402480315918662210015589253, 3.23698888432409870110642165395, 3.35611019471545829189951929903, 3.66843940760103763017763795582, 3.69642135781798168197581543462, 3.70029420914446370702396553005
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.