Dirichlet series
| L(s) = 1 | − 3·3-s − 4·5-s + 10·7-s − 6·9-s + 10·11-s − 10·13-s + 12·15-s − 11·17-s + 2·19-s − 30·21-s − 8·23-s − 16·25-s + 25·27-s − 8·29-s − 23·31-s − 30·33-s − 40·35-s + 10·37-s + 30·39-s − 18·41-s + 12·43-s + 24·45-s − 36·47-s + 55·49-s + 33·51-s − 21·53-s − 40·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s + 3.77·7-s − 2·9-s + 3.01·11-s − 2.77·13-s + 3.09·15-s − 2.66·17-s + 0.458·19-s − 6.54·21-s − 1.66·23-s − 3.19·25-s + 4.81·27-s − 1.48·29-s − 4.13·31-s − 5.22·33-s − 6.76·35-s + 1.64·37-s + 4.80·39-s − 2.81·41-s + 1.82·43-s + 3.57·45-s − 5.25·47-s + 55/7·49-s + 4.62·51-s − 2.88·53-s − 5.39·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.14290\times 10^{18}\) |
| Root analytic conductor: | \(7.99651\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
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 | \( 1 \) |
| 7 | \( ( 1 - T )^{10} \) | |
| 11 | \( ( 1 - T )^{10} \) | |
| 13 | \( ( 1 + T )^{10} \) | |
| good | 3 | \( 1 + p T + 5 p T^{2} + 38 T^{3} + 37 p T^{4} + 242 T^{5} + 521 T^{6} + 38 p^{3} T^{7} + 605 p T^{8} + 3421 T^{9} + 5570 T^{10} + 3421 p T^{11} + 605 p^{3} T^{12} + 38 p^{6} T^{13} + 521 p^{4} T^{14} + 242 p^{5} T^{15} + 37 p^{7} T^{16} + 38 p^{7} T^{17} + 5 p^{9} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + 4 T + 32 T^{2} + 98 T^{3} + 452 T^{4} + 1161 T^{5} + 3978 T^{6} + 9108 T^{7} + 25908 T^{8} + 54954 T^{9} + 139448 T^{10} + 54954 p T^{11} + 25908 p^{2} T^{12} + 9108 p^{3} T^{13} + 3978 p^{4} T^{14} + 1161 p^{5} T^{15} + 452 p^{6} T^{16} + 98 p^{7} T^{17} + 32 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 11 T + 165 T^{2} + 1334 T^{3} + 11945 T^{4} + 76914 T^{5} + 515157 T^{6} + 161290 p T^{7} + 14828041 T^{8} + 66343271 T^{9} + 299048318 T^{10} + 66343271 p T^{11} + 14828041 p^{2} T^{12} + 161290 p^{4} T^{13} + 515157 p^{4} T^{14} + 76914 p^{5} T^{15} + 11945 p^{6} T^{16} + 1334 p^{7} T^{17} + 165 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 2 T + 106 T^{2} - 364 T^{3} + 5898 T^{4} - 23557 T^{5} + 232308 T^{6} - 889420 T^{7} + 6771210 T^{8} - 23334494 T^{9} + 147824662 T^{10} - 23334494 p T^{11} + 6771210 p^{2} T^{12} - 889420 p^{3} T^{13} + 232308 p^{4} T^{14} - 23557 p^{5} T^{15} + 5898 p^{6} T^{16} - 364 p^{7} T^{17} + 106 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 8 T + 173 T^{2} + 1108 T^{3} + 14006 T^{4} + 3299 p T^{5} + 715666 T^{6} + 3348305 T^{7} + 25743697 T^{8} + 4553868 p T^{9} + 684091730 T^{10} + 4553868 p^{2} T^{11} + 25743697 p^{2} T^{12} + 3348305 p^{3} T^{13} + 715666 p^{4} T^{14} + 3299 p^{6} T^{15} + 14006 p^{6} T^{16} + 1108 p^{7} T^{17} + 173 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 8 T + 161 T^{2} + 968 T^{3} + 12066 T^{4} + 58869 T^{5} + 584656 T^{6} + 2397855 T^{7} + 21304721 T^{8} + 77493084 T^{9} + 656370238 T^{10} + 77493084 p T^{11} + 21304721 p^{2} T^{12} + 2397855 p^{3} T^{13} + 584656 p^{4} T^{14} + 58869 p^{5} T^{15} + 12066 p^{6} T^{16} + 968 p^{7} T^{17} + 161 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 23 T + 444 T^{2} + 5927 T^{3} + 69838 T^{4} + 680889 T^{5} + 6017694 T^{6} + 46520575 T^{7} + 330538885 T^{8} + 2099194338 T^{9} + 12325983924 T^{10} + 2099194338 p T^{11} + 330538885 p^{2} T^{12} + 46520575 p^{3} T^{13} + 6017694 p^{4} T^{14} + 680889 p^{5} T^{15} + 69838 p^{6} T^{16} + 5927 p^{7} T^{17} + 444 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 10 T + 220 T^{2} - 1553 T^{3} + 21750 T^{4} - 122189 T^{5} + 1420560 T^{6} - 6947203 T^{7} + 72917701 T^{8} - 321967157 T^{9} + 3020323856 T^{10} - 321967157 p T^{11} + 72917701 p^{2} T^{12} - 6947203 p^{3} T^{13} + 1420560 p^{4} T^{14} - 122189 p^{5} T^{15} + 21750 p^{6} T^{16} - 1553 p^{7} T^{17} + 220 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 18 T + 238 T^{2} + 2393 T^{3} + 20962 T^{4} + 161885 T^{5} + 1160836 T^{6} + 8080847 T^{7} + 56310233 T^{8} + 374883769 T^{9} + 2462477380 T^{10} + 374883769 p T^{11} + 56310233 p^{2} T^{12} + 8080847 p^{3} T^{13} + 1160836 p^{4} T^{14} + 161885 p^{5} T^{15} + 20962 p^{6} T^{16} + 2393 p^{7} T^{17} + 238 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 12 T + 363 T^{2} - 3490 T^{3} + 59662 T^{4} - 477863 T^{5} + 5989474 T^{6} - 40894050 T^{7} + 413451091 T^{8} - 2430753754 T^{9} + 20706867338 T^{10} - 2430753754 p T^{11} + 413451091 p^{2} T^{12} - 40894050 p^{3} T^{13} + 5989474 p^{4} T^{14} - 477863 p^{5} T^{15} + 59662 p^{6} T^{16} - 3490 p^{7} T^{17} + 363 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 36 T + 813 T^{2} + 13040 T^{3} + 170094 T^{4} + 1865695 T^{5} + 18186994 T^{6} + 159801787 T^{7} + 1302432025 T^{8} + 9841123220 T^{9} + 69881745858 T^{10} + 9841123220 p T^{11} + 1302432025 p^{2} T^{12} + 159801787 p^{3} T^{13} + 18186994 p^{4} T^{14} + 1865695 p^{5} T^{15} + 170094 p^{6} T^{16} + 13040 p^{7} T^{17} + 813 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 21 T + 458 T^{2} + 5917 T^{3} + 1432 p T^{4} + 742237 T^{5} + 7436742 T^{6} + 63387885 T^{7} + 560781624 T^{8} + 4336221081 T^{9} + 33872463384 T^{10} + 4336221081 p T^{11} + 560781624 p^{2} T^{12} + 63387885 p^{3} T^{13} + 7436742 p^{4} T^{14} + 742237 p^{5} T^{15} + 1432 p^{7} T^{16} + 5917 p^{7} T^{17} + 458 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 13 T + 425 T^{2} + 4830 T^{3} + 83498 T^{4} + 837647 T^{5} + 10216944 T^{6} + 91417039 T^{7} + 889363057 T^{8} + 7127736941 T^{9} + 59241940758 T^{10} + 7127736941 p T^{11} + 889363057 p^{2} T^{12} + 91417039 p^{3} T^{13} + 10216944 p^{4} T^{14} + 837647 p^{5} T^{15} + 83498 p^{6} T^{16} + 4830 p^{7} T^{17} + 425 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 2 T + 269 T^{2} + 985 T^{3} + 39081 T^{4} + 178586 T^{5} + 4174189 T^{6} + 19263885 T^{7} + 354834971 T^{8} + 1492227926 T^{9} + 24194541034 T^{10} + 1492227926 p T^{11} + 354834971 p^{2} T^{12} + 19263885 p^{3} T^{13} + 4174189 p^{4} T^{14} + 178586 p^{5} T^{15} + 39081 p^{6} T^{16} + 985 p^{7} T^{17} + 269 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 4 T + 361 T^{2} + 974 T^{3} + 53789 T^{4} + 52384 T^{5} + 4197223 T^{6} - 8053586 T^{7} + 184733463 T^{8} - 1469351166 T^{9} + 7578251062 T^{10} - 1469351166 p T^{11} + 184733463 p^{2} T^{12} - 8053586 p^{3} T^{13} + 4197223 p^{4} T^{14} + 52384 p^{5} T^{15} + 53789 p^{6} T^{16} + 974 p^{7} T^{17} + 361 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 24 T + 693 T^{2} + 11236 T^{3} + 2790 p T^{4} + 2528474 T^{5} + 33949518 T^{6} + 359734038 T^{7} + 3953042517 T^{8} + 35526391064 T^{9} + 329357805242 T^{10} + 35526391064 p T^{11} + 3953042517 p^{2} T^{12} + 359734038 p^{3} T^{13} + 33949518 p^{4} T^{14} + 2528474 p^{5} T^{15} + 2790 p^{7} T^{16} + 11236 p^{7} T^{17} + 693 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 23 T + 676 T^{2} + 11028 T^{3} + 188109 T^{4} + 2401330 T^{5} + 30299132 T^{6} + 321664326 T^{7} + 3320214618 T^{8} + 30614873121 T^{9} + 273932980024 T^{10} + 30614873121 p T^{11} + 3320214618 p^{2} T^{12} + 321664326 p^{3} T^{13} + 30299132 p^{4} T^{14} + 2401330 p^{5} T^{15} + 188109 p^{6} T^{16} + 11028 p^{7} T^{17} + 676 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 14 T + 397 T^{2} - 5082 T^{3} + 94296 T^{4} - 1053191 T^{5} + 14985812 T^{6} - 148481456 T^{7} + 1777691157 T^{8} - 15481851080 T^{9} + 159375689706 T^{10} - 15481851080 p T^{11} + 1777691157 p^{2} T^{12} - 148481456 p^{3} T^{13} + 14985812 p^{4} T^{14} - 1053191 p^{5} T^{15} + 94296 p^{6} T^{16} - 5082 p^{7} T^{17} + 397 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 9 T + 465 T^{2} + 4111 T^{3} + 110304 T^{4} + 956241 T^{5} + 17536156 T^{6} + 145998105 T^{7} + 2079412063 T^{8} + 16132314327 T^{9} + 193464771898 T^{10} + 16132314327 p T^{11} + 2079412063 p^{2} T^{12} + 145998105 p^{3} T^{13} + 17536156 p^{4} T^{14} + 956241 p^{5} T^{15} + 110304 p^{6} T^{16} + 4111 p^{7} T^{17} + 465 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 18 T + 498 T^{2} + 6870 T^{3} + 114090 T^{4} + 1249991 T^{5} + 16070698 T^{6} + 144657182 T^{7} + 1627247754 T^{8} + 13217567006 T^{9} + 145173992544 T^{10} + 13217567006 p T^{11} + 1627247754 p^{2} T^{12} + 144657182 p^{3} T^{13} + 16070698 p^{4} T^{14} + 1249991 p^{5} T^{15} + 114090 p^{6} T^{16} + 6870 p^{7} T^{17} + 498 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 9 T + 686 T^{2} + 5198 T^{3} + 225881 T^{4} + 1477486 T^{5} + 47611206 T^{6} + 272379076 T^{7} + 7147461446 T^{8} + 35836368167 T^{9} + 798742143352 T^{10} + 35836368167 p T^{11} + 7147461446 p^{2} T^{12} + 272379076 p^{3} T^{13} + 47611206 p^{4} T^{14} + 1477486 p^{5} T^{15} + 225881 p^{6} T^{16} + 5198 p^{7} T^{17} + 686 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.20393409460861109051946856417, −3.05296306686259236472760729233, −2.96837375855780339059216085635, −2.91071508483435353946840950260, −2.87123695096230459548804518693, −2.40749121242903071054626422604, −2.36689248531105091327555389031, −2.26863028620679507990125190940, −2.23148614075677518784901944223, −2.20797236753581802913112864679, −2.17597630756953008047448112537, −2.14694779016937031932130200749, −2.07328041605003480890090777344, −1.99967597027896640899113808970, −1.82072693291509366629794160559, −1.74000174446129118102766906692, −1.59666618904531843824602430645, −1.43407226768454451649766107982, −1.40134110116748626878017128393, −1.34391270135139237027720127639, −1.33458166745723946187192988564, −1.10644961217704634406813839228, −1.08971616204818068547373109392, −1.00109027818303758658120781150, −0.813127353504909593044552564942, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.813127353504909593044552564942, 1.00109027818303758658120781150, 1.08971616204818068547373109392, 1.10644961217704634406813839228, 1.33458166745723946187192988564, 1.34391270135139237027720127639, 1.40134110116748626878017128393, 1.43407226768454451649766107982, 1.59666618904531843824602430645, 1.74000174446129118102766906692, 1.82072693291509366629794160559, 1.99967597027896640899113808970, 2.07328041605003480890090777344, 2.14694779016937031932130200749, 2.17597630756953008047448112537, 2.20797236753581802913112864679, 2.23148614075677518784901944223, 2.26863028620679507990125190940, 2.36689248531105091327555389031, 2.40749121242903071054626422604, 2.87123695096230459548804518693, 2.91071508483435353946840950260, 2.96837375855780339059216085635, 3.05296306686259236472760729233, 3.20393409460861109051946856417
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.