Dirichlet series
| L(s) = 1 | + 5·2-s + 3-s + 13·4-s + 5-s + 5·6-s + 5·7-s + 22·8-s + 3·9-s + 5·10-s + 8·11-s + 13·12-s + 25·14-s + 15-s + 33·16-s − 23·17-s + 15·18-s − 13·19-s + 13·20-s + 5·21-s + 40·22-s − 21·23-s + 22·24-s + 65·28-s + 2·29-s + 5·30-s − 20·31-s + 55·32-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 0.577·3-s + 13/2·4-s + 0.447·5-s + 2.04·6-s + 1.88·7-s + 7.77·8-s + 9-s + 1.58·10-s + 2.41·11-s + 3.75·12-s + 6.68·14-s + 0.258·15-s + 33/4·16-s − 5.57·17-s + 3.53·18-s − 2.98·19-s + 2.90·20-s + 1.09·21-s + 8.52·22-s − 4.37·23-s + 4.49·24-s + 12.2·28-s + 0.371·29-s + 0.912·30-s − 3.59·31-s + 9.72·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(5^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.426333\) |
| Root analytic conductor: | \(0.958269\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 5^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.52871680\) |
| \(L(\frac12)\) | \(\approx\) | \(10.52871680\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 23 | \( 1 + 21 T + 243 T^{2} + 1913 T^{3} + 11771 T^{4} + 60543 T^{5} + 11771 p T^{6} + 1913 p^{2} T^{7} + 243 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 - 5 T + 3 p^{2} T^{2} - 17 T^{3} + 3 p T^{4} + 37 T^{5} - 49 p T^{6} + 119 T^{7} - 25 T^{8} - 53 p^{2} T^{9} + 461 T^{10} - 53 p^{3} T^{11} - 25 p^{2} T^{12} + 119 p^{3} T^{13} - 49 p^{5} T^{14} + 37 p^{5} T^{15} + 3 p^{7} T^{16} - 17 p^{7} T^{17} + 3 p^{10} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) |
| 3 | \( ( 1 - 2 p T + 14 T^{2} - 7 T^{3} - 46 T^{4} + 133 T^{5} - 46 p T^{6} - 7 p^{2} T^{7} + 14 p^{3} T^{8} - 2 p^{5} T^{9} + p^{5} T^{10} )( 1 + 5 T + 14 T^{2} + 26 T^{3} + 14 p T^{4} + 67 T^{5} + 14 p^{2} T^{6} + 26 p^{2} T^{7} + 14 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} ) \) | |
| 7 | \( 1 - 5 T + p T^{2} - 22 T^{3} + 61 T^{4} - 9 p T^{5} + 262 T^{6} + 704 T^{7} - 3165 T^{8} + 7828 T^{9} - 38149 T^{10} + 7828 p T^{11} - 3165 p^{2} T^{12} + 704 p^{3} T^{13} + 262 p^{4} T^{14} - 9 p^{6} T^{15} + 61 p^{6} T^{16} - 22 p^{7} T^{17} + p^{9} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 8 T + 20 T^{2} - 116 T^{3} + 785 T^{4} - 1924 T^{5} + 7648 T^{6} - 42781 T^{7} + 84254 T^{8} - 306621 T^{9} + 1799073 T^{10} - 306621 p T^{11} + 84254 p^{2} T^{12} - 42781 p^{3} T^{13} + 7648 p^{4} T^{14} - 1924 p^{5} T^{15} + 785 p^{6} T^{16} - 116 p^{7} T^{17} + 20 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 2 T^{2} + 11 T^{3} + p^{2} T^{4} + 407 T^{5} + 1785 T^{6} - 2068 T^{7} + 30475 T^{8} + 77154 T^{9} + 232145 T^{10} + 77154 p T^{11} + 30475 p^{2} T^{12} - 2068 p^{3} T^{13} + 1785 p^{4} T^{14} + 407 p^{5} T^{15} + p^{8} T^{16} + 11 p^{7} T^{17} - 2 p^{8} T^{18} + p^{10} T^{20} \) | |
| 17 | \( 1 + 23 T + 248 T^{2} + 1793 T^{3} + 10843 T^{4} + 61388 T^{5} + 327661 T^{6} + 1644533 T^{7} + 7815982 T^{8} + 35008785 T^{9} + 148068887 T^{10} + 35008785 p T^{11} + 7815982 p^{2} T^{12} + 1644533 p^{3} T^{13} + 327661 p^{4} T^{14} + 61388 p^{5} T^{15} + 10843 p^{6} T^{16} + 1793 p^{7} T^{17} + 248 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 13 T + 29 T^{2} - 409 T^{3} - 123 p T^{4} + 5165 T^{5} + 66426 T^{6} + 139 p T^{7} - 1149892 T^{8} - 297132 T^{9} + 19829679 T^{10} - 297132 p T^{11} - 1149892 p^{2} T^{12} + 139 p^{4} T^{13} + 66426 p^{4} T^{14} + 5165 p^{5} T^{15} - 123 p^{7} T^{16} - 409 p^{7} T^{17} + 29 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 2 T + 41 T^{2} - 222 T^{3} + 531 T^{4} - 9408 T^{5} + 47813 T^{6} - 301118 T^{7} + 2010297 T^{8} - 10226854 T^{9} + 44968155 T^{10} - 10226854 p T^{11} + 2010297 p^{2} T^{12} - 301118 p^{3} T^{13} + 47813 p^{4} T^{14} - 9408 p^{5} T^{15} + 531 p^{6} T^{16} - 222 p^{7} T^{17} + 41 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 20 T + 127 T^{2} + 127 T^{3} - 1089 T^{4} - 3794 T^{5} - 73933 T^{6} - 716061 T^{7} - 2339386 T^{8} + 4043930 T^{9} + 65396035 T^{10} + 4043930 p T^{11} - 2339386 p^{2} T^{12} - 716061 p^{3} T^{13} - 73933 p^{4} T^{14} - 3794 p^{5} T^{15} - 1089 p^{6} T^{16} + 127 p^{7} T^{17} + 127 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 13 T + 132 T^{2} - 1235 T^{3} + 11171 T^{4} - 85668 T^{5} + 658381 T^{6} - 4704949 T^{7} + 33057222 T^{8} - 212975933 T^{9} + 1343462603 T^{10} - 212975933 p T^{11} + 33057222 p^{2} T^{12} - 4704949 p^{3} T^{13} + 658381 p^{4} T^{14} - 85668 p^{5} T^{15} + 11171 p^{6} T^{16} - 1235 p^{7} T^{17} + 132 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 5 T + 50 T^{2} - 681 T^{3} - 2573 T^{4} - 37436 T^{5} + 238919 T^{6} + 364515 T^{7} + 12937390 T^{8} - 71985111 T^{9} + 8447451 T^{10} - 71985111 p T^{11} + 12937390 p^{2} T^{12} + 364515 p^{3} T^{13} + 238919 p^{4} T^{14} - 37436 p^{5} T^{15} - 2573 p^{6} T^{16} - 681 p^{7} T^{17} + 50 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 15 T + 204 T^{2} - 2129 T^{3} + 19643 T^{4} - 172738 T^{5} + 1385269 T^{6} - 10775981 T^{7} + 77916158 T^{8} - 539582197 T^{9} + 3599056165 T^{10} - 539582197 p T^{11} + 77916158 p^{2} T^{12} - 10775981 p^{3} T^{13} + 1385269 p^{4} T^{14} - 172738 p^{5} T^{15} + 19643 p^{6} T^{16} - 2129 p^{7} T^{17} + 204 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 26 T + 424 T^{2} + 4806 T^{3} + 43985 T^{4} + 327085 T^{5} + 43985 p T^{6} + 4806 p^{2} T^{7} + 424 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 - 6 T + 126 T^{2} - 1670 T^{3} + 18434 T^{4} - 163092 T^{5} + 1868723 T^{6} - 15540916 T^{7} + 130470814 T^{8} - 1013135108 T^{9} + 8246457637 T^{10} - 1013135108 p T^{11} + 130470814 p^{2} T^{12} - 15540916 p^{3} T^{13} + 1868723 p^{4} T^{14} - 163092 p^{5} T^{15} + 18434 p^{6} T^{16} - 1670 p^{7} T^{17} + 126 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 5 T + 142 T^{2} - 646 T^{3} + 8536 T^{4} - 26742 T^{5} + 287886 T^{6} + 702648 T^{7} + 10262887 T^{8} + 139977013 T^{9} + 674338952 T^{10} + 139977013 p T^{11} + 10262887 p^{2} T^{12} + 702648 p^{3} T^{13} + 287886 p^{4} T^{14} - 26742 p^{5} T^{15} + 8536 p^{6} T^{16} - 646 p^{7} T^{17} + 142 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 3 T + 14 T^{2} - 174 T^{3} + 3200 T^{4} + 46207 T^{5} + 124734 T^{6} - 489978 T^{7} + 801426 T^{8} + 740024 p T^{9} + 1300242855 T^{10} + 740024 p^{2} T^{11} + 801426 p^{2} T^{12} - 489978 p^{3} T^{13} + 124734 p^{4} T^{14} + 46207 p^{5} T^{15} + 3200 p^{6} T^{16} - 174 p^{7} T^{17} + 14 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 20 T + 223 T^{2} - 1822 T^{3} + 4262 T^{4} + 78106 T^{5} - 1241112 T^{6} + 7658230 T^{7} + 22056296 T^{8} - 902388884 T^{9} + 10137910717 T^{10} - 902388884 p T^{11} + 22056296 p^{2} T^{12} + 7658230 p^{3} T^{13} - 1241112 p^{4} T^{14} + 78106 p^{5} T^{15} + 4262 p^{6} T^{16} - 1822 p^{7} T^{17} + 223 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 22 T + 204 T^{2} + 209 T^{3} - 1361 T^{4} + 99759 T^{5} + 1632143 T^{6} + 4544078 T^{7} - 8138253 T^{8} + 377062070 T^{9} + 8288280417 T^{10} + 377062070 p T^{11} - 8138253 p^{2} T^{12} + 4544078 p^{3} T^{13} + 1632143 p^{4} T^{14} + 99759 p^{5} T^{15} - 1361 p^{6} T^{16} + 209 p^{7} T^{17} + 204 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 43 T + 797 T^{2} - 8021 T^{3} + 47021 T^{4} - 205129 T^{5} + 1802674 T^{6} - 24182707 T^{7} + 240208550 T^{8} - 1756457458 T^{9} + 12854913015 T^{10} - 1756457458 p T^{11} + 240208550 p^{2} T^{12} - 24182707 p^{3} T^{13} + 1802674 p^{4} T^{14} - 205129 p^{5} T^{15} + 47021 p^{6} T^{16} - 8021 p^{7} T^{17} + 797 p^{8} T^{18} - 43 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 51 T + 1114 T^{2} - 13295 T^{3} + 92135 T^{4} - 385584 T^{5} + 1188383 T^{6} - 2740799 T^{7} + 64216836 T^{8} - 2455180103 T^{9} + 33339862885 T^{10} - 2455180103 p T^{11} + 64216836 p^{2} T^{12} - 2740799 p^{3} T^{13} + 1188383 p^{4} T^{14} - 385584 p^{5} T^{15} + 92135 p^{6} T^{16} - 13295 p^{7} T^{17} + 1114 p^{8} T^{18} - 51 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 17 T - 47 T^{2} - 2793 T^{3} - 1329 T^{4} + 469277 T^{5} + 3476090 T^{6} - 30330137 T^{7} - 432941786 T^{8} + 1652517680 T^{9} + 51926903887 T^{10} + 1652517680 p T^{11} - 432941786 p^{2} T^{12} - 30330137 p^{3} T^{13} + 3476090 p^{4} T^{14} + 469277 p^{5} T^{15} - 1329 p^{6} T^{16} - 2793 p^{7} T^{17} - 47 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 57 T + 1455 T^{2} - 22697 T^{3} + 269934 T^{4} - 3046104 T^{5} + 33584052 T^{6} - 328175911 T^{7} + 3078564586 T^{8} - 32141776413 T^{9} + 327813403533 T^{10} - 32141776413 p T^{11} + 3078564586 p^{2} T^{12} - 328175911 p^{3} T^{13} + 33584052 p^{4} T^{14} - 3046104 p^{5} T^{15} + 269934 p^{6} T^{16} - 22697 p^{7} T^{17} + 1455 p^{8} T^{18} - 57 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 52 T + 1221 T^{2} - 17946 T^{3} + 192969 T^{4} - 1571526 T^{5} + 80557 p T^{6} + 34957602 T^{7} - 1703623889 T^{8} + 28959932998 T^{9} - 334806911165 T^{10} + 28959932998 p T^{11} - 1703623889 p^{2} T^{12} + 34957602 p^{3} T^{13} + 80557 p^{5} T^{14} - 1571526 p^{5} T^{15} + 192969 p^{6} T^{16} - 17946 p^{7} T^{17} + 1221 p^{8} T^{18} - 52 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−5.36043344164897587173746029547, −5.33233567324296098300253193928, −5.21203818388439002762225678670, −4.92260337106200788416613313347, −4.87178539010652911212605913500, −4.82827373573867513169421665175, −4.50466522741825984773697685515, −4.44375388383588924776995605561, −4.43469900075623912426136651106, −4.20981357212134820057600913110, −4.03553998627969204557224569917, −3.92461660155874049383890990899, −3.88601739345142323769478183359, −3.78341123000989059494978050425, −3.57312716047285884889582544714, −3.42794605069091653498020812561, −3.28178555272531201854561675266, −2.94203416129092519818582928257, −2.39924828180303328060907750517, −2.22148551717071246727612662844, −2.15578584535344603299560178573, −2.08479681043746701826867835677, −1.95300775100368514876716889292, −1.85001603054626592117685022471, −1.56374161847528184207976961542, 1.56374161847528184207976961542, 1.85001603054626592117685022471, 1.95300775100368514876716889292, 2.08479681043746701826867835677, 2.15578584535344603299560178573, 2.22148551717071246727612662844, 2.39924828180303328060907750517, 2.94203416129092519818582928257, 3.28178555272531201854561675266, 3.42794605069091653498020812561, 3.57312716047285884889582544714, 3.78341123000989059494978050425, 3.88601739345142323769478183359, 3.92461660155874049383890990899, 4.03553998627969204557224569917, 4.20981357212134820057600913110, 4.43469900075623912426136651106, 4.44375388383588924776995605561, 4.50466522741825984773697685515, 4.82827373573867513169421665175, 4.87178539010652911212605913500, 4.92260337106200788416613313347, 5.21203818388439002762225678670, 5.33233567324296098300253193928, 5.36043344164897587173746029547
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.