Dirichlet series
| L(s) = 1 | + 3·2-s + 13·3-s + 39·6-s − 4·7-s − 6·8-s + 91·9-s + 16·11-s − 8·13-s − 12·14-s + 16-s + 12·17-s + 273·18-s + 28·19-s − 52·21-s + 48·22-s + 6·23-s − 78·24-s − 24·26-s + 455·27-s + 12·29-s + 26·31-s + 13·32-s + 208·33-s + 36·34-s − 4·37-s + 84·38-s − 104·39-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 7.50·3-s + 15.9·6-s − 1.51·7-s − 2.12·8-s + 91/3·9-s + 4.82·11-s − 2.21·13-s − 3.20·14-s + 1/4·16-s + 2.91·17-s + 64.3·18-s + 6.42·19-s − 11.3·21-s + 10.2·22-s + 1.25·23-s − 15.9·24-s − 4.70·26-s + 87.5·27-s + 2.22·29-s + 4.66·31-s + 2.29·32-s + 36.2·33-s + 6.17·34-s − 0.657·37-s + 13.6·38-s − 16.6·39-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{13} \cdot 5^{26} \cdot 47^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{13} \cdot 5^{26} \cdot 47^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(3^{13} \cdot 5^{26} \cdot 47^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.96098\times 10^{18}\) |
| Root analytic conductor: | \(5.30539\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 3^{13} \cdot 5^{26} \cdot 47^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(66280.76860\) |
| \(L(\frac12)\) | \(\approx\) | \(66280.76860\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{13} \) |
| 5 | \( 1 \) | |
| 47 | \( ( 1 - T )^{13} \) | |
| good | 2 | \( 1 - 3 T + 9 T^{2} - 21 T^{3} + 11 p^{2} T^{4} - 11 p^{3} T^{5} + 21 p^{3} T^{6} - 19 p^{4} T^{7} + 541 T^{8} - 891 T^{9} + 1447 T^{10} - 2211 T^{11} + 1641 p T^{12} - 2363 p T^{13} + 1641 p^{2} T^{14} - 2211 p^{2} T^{15} + 1447 p^{3} T^{16} - 891 p^{4} T^{17} + 541 p^{5} T^{18} - 19 p^{10} T^{19} + 21 p^{10} T^{20} - 11 p^{11} T^{21} + 11 p^{11} T^{22} - 21 p^{10} T^{23} + 9 p^{11} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) |
| 7 | \( 1 + 4 T + 43 T^{2} + 164 T^{3} + 968 T^{4} + 3284 T^{5} + 14512 T^{6} + 43728 T^{7} + 23256 p T^{8} + 444540 T^{9} + 211688 p T^{10} + 537652 p T^{11} + 11636362 T^{12} + 27900056 T^{13} + 11636362 p T^{14} + 537652 p^{3} T^{15} + 211688 p^{4} T^{16} + 444540 p^{4} T^{17} + 23256 p^{6} T^{18} + 43728 p^{6} T^{19} + 14512 p^{7} T^{20} + 3284 p^{8} T^{21} + 968 p^{9} T^{22} + 164 p^{10} T^{23} + 43 p^{11} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 11 | \( 1 - 16 T + 160 T^{2} - 1220 T^{3} + 7947 T^{4} - 45632 T^{5} + 237101 T^{6} - 1129996 T^{7} + 5012726 T^{8} - 20843168 T^{9} + 81857995 T^{10} - 305033552 T^{11} + 1083982374 T^{12} - 3677270560 T^{13} + 1083982374 p T^{14} - 305033552 p^{2} T^{15} + 81857995 p^{3} T^{16} - 20843168 p^{4} T^{17} + 5012726 p^{5} T^{18} - 1129996 p^{6} T^{19} + 237101 p^{7} T^{20} - 45632 p^{8} T^{21} + 7947 p^{9} T^{22} - 1220 p^{10} T^{23} + 160 p^{11} T^{24} - 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 13 | \( 1 + 8 T + 8 p T^{2} + 586 T^{3} + 4658 T^{4} + 20634 T^{5} + 127163 T^{6} + 457370 T^{7} + 2417555 T^{8} + 7153212 T^{9} + 35398956 T^{10} + 89193276 T^{11} + 455325583 T^{12} + 1089302180 T^{13} + 455325583 p T^{14} + 89193276 p^{2} T^{15} + 35398956 p^{3} T^{16} + 7153212 p^{4} T^{17} + 2417555 p^{5} T^{18} + 457370 p^{6} T^{19} + 127163 p^{7} T^{20} + 20634 p^{8} T^{21} + 4658 p^{9} T^{22} + 586 p^{10} T^{23} + 8 p^{12} T^{24} + 8 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 - 12 T + 172 T^{2} - 1494 T^{3} + 13174 T^{4} - 90694 T^{5} + 616875 T^{6} - 3565606 T^{7} + 20313823 T^{8} - 102406720 T^{9} + 511482204 T^{10} - 2311763380 T^{11} + 10416774951 T^{12} - 42958099964 T^{13} + 10416774951 p T^{14} - 2311763380 p^{2} T^{15} + 511482204 p^{3} T^{16} - 102406720 p^{4} T^{17} + 20313823 p^{5} T^{18} - 3565606 p^{6} T^{19} + 616875 p^{7} T^{20} - 90694 p^{8} T^{21} + 13174 p^{9} T^{22} - 1494 p^{10} T^{23} + 172 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 - 28 T + 492 T^{2} - 6528 T^{3} + 3760 p T^{4} - 671582 T^{5} + 5570571 T^{6} - 41447750 T^{7} + 279945741 T^{8} - 1730173760 T^{9} + 9843617662 T^{10} - 51759697610 T^{11} + 13275742313 p T^{12} - 1140915091724 T^{13} + 13275742313 p^{2} T^{14} - 51759697610 p^{2} T^{15} + 9843617662 p^{3} T^{16} - 1730173760 p^{4} T^{17} + 279945741 p^{5} T^{18} - 41447750 p^{6} T^{19} + 5570571 p^{7} T^{20} - 671582 p^{8} T^{21} + 3760 p^{10} T^{22} - 6528 p^{10} T^{23} + 492 p^{11} T^{24} - 28 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 - 6 T + 179 T^{2} - 1110 T^{3} + 16520 T^{4} - 99326 T^{5} + 1031960 T^{6} - 5774728 T^{7} + 47919376 T^{8} - 245166826 T^{9} + 1726202744 T^{10} - 8022590450 T^{11} + 49395875746 T^{12} - 207096226820 T^{13} + 49395875746 p T^{14} - 8022590450 p^{2} T^{15} + 1726202744 p^{3} T^{16} - 245166826 p^{4} T^{17} + 47919376 p^{5} T^{18} - 5774728 p^{6} T^{19} + 1031960 p^{7} T^{20} - 99326 p^{8} T^{21} + 16520 p^{9} T^{22} - 1110 p^{10} T^{23} + 179 p^{11} T^{24} - 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 - 12 T + 353 T^{2} - 3634 T^{3} + 58064 T^{4} - 516274 T^{5} + 5892988 T^{6} - 45548688 T^{7} + 412051420 T^{8} - 2782113662 T^{9} + 723505064 p T^{10} - 124034022222 T^{11} + 27624127626 p T^{12} - 4140729684776 T^{13} + 27624127626 p^{2} T^{14} - 124034022222 p^{2} T^{15} + 723505064 p^{4} T^{16} - 2782113662 p^{4} T^{17} + 412051420 p^{5} T^{18} - 45548688 p^{6} T^{19} + 5892988 p^{7} T^{20} - 516274 p^{8} T^{21} + 58064 p^{9} T^{22} - 3634 p^{10} T^{23} + 353 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 - 26 T + 615 T^{2} - 9788 T^{3} + 140498 T^{4} - 1655532 T^{5} + 17842530 T^{6} - 168349068 T^{7} + 1469428579 T^{8} - 11552018806 T^{9} + 84641925597 T^{10} - 566329743288 T^{11} + 3546285536368 T^{12} - 20396495022600 T^{13} + 3546285536368 p T^{14} - 566329743288 p^{2} T^{15} + 84641925597 p^{3} T^{16} - 11552018806 p^{4} T^{17} + 1469428579 p^{5} T^{18} - 168349068 p^{6} T^{19} + 17842530 p^{7} T^{20} - 1655532 p^{8} T^{21} + 140498 p^{9} T^{22} - 9788 p^{10} T^{23} + 615 p^{11} T^{24} - 26 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 + 4 T + 234 T^{2} + 16 p T^{3} + 27669 T^{4} + 41212 T^{5} + 2205517 T^{6} + 1260768 T^{7} + 132439392 T^{8} - 27845892 T^{9} + 6457826825 T^{10} - 4860396528 T^{11} + 269724777434 T^{12} - 248212944760 T^{13} + 269724777434 p T^{14} - 4860396528 p^{2} T^{15} + 6457826825 p^{3} T^{16} - 27845892 p^{4} T^{17} + 132439392 p^{5} T^{18} + 1260768 p^{6} T^{19} + 2205517 p^{7} T^{20} + 41212 p^{8} T^{21} + 27669 p^{9} T^{22} + 16 p^{11} T^{23} + 234 p^{11} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 24 T + 612 T^{2} - 9822 T^{3} + 150038 T^{4} - 1847180 T^{5} + 21288067 T^{6} - 215241016 T^{7} + 2037250777 T^{8} - 17576208548 T^{9} + 142293233694 T^{10} - 1069300527146 T^{11} + 7551786779731 T^{12} - 49869177957488 T^{13} + 7551786779731 p T^{14} - 1069300527146 p^{2} T^{15} + 142293233694 p^{3} T^{16} - 17576208548 p^{4} T^{17} + 2037250777 p^{5} T^{18} - 215241016 p^{6} T^{19} + 21288067 p^{7} T^{20} - 1847180 p^{8} T^{21} + 150038 p^{9} T^{22} - 9822 p^{10} T^{23} + 612 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 + 6 T + 349 T^{2} + 1944 T^{3} + 61308 T^{4} + 317852 T^{5} + 7139868 T^{6} + 34392792 T^{7} + 613040061 T^{8} + 2729956218 T^{9} + 40889921529 T^{10} + 166966894544 T^{11} + 2176927965424 T^{12} + 8055386171144 T^{13} + 2176927965424 p T^{14} + 166966894544 p^{2} T^{15} + 40889921529 p^{3} T^{16} + 2729956218 p^{4} T^{17} + 613040061 p^{5} T^{18} + 34392792 p^{6} T^{19} + 7139868 p^{7} T^{20} + 317852 p^{8} T^{21} + 61308 p^{9} T^{22} + 1944 p^{10} T^{23} + 349 p^{11} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 - 6 T + 522 T^{2} - 2894 T^{3} + 131474 T^{4} - 679572 T^{5} + 21193043 T^{6} - 102091710 T^{7} + 2440327541 T^{8} - 10876352270 T^{9} + 211992447760 T^{10} - 862986663604 T^{11} + 14295033685919 T^{12} - 52201671725264 T^{13} + 14295033685919 p T^{14} - 862986663604 p^{2} T^{15} + 211992447760 p^{3} T^{16} - 10876352270 p^{4} T^{17} + 2440327541 p^{5} T^{18} - 102091710 p^{6} T^{19} + 21193043 p^{7} T^{20} - 679572 p^{8} T^{21} + 131474 p^{9} T^{22} - 2894 p^{10} T^{23} + 522 p^{11} T^{24} - 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 - 34 T + 1126 T^{2} - 24354 T^{3} + 487840 T^{4} - 7904032 T^{5} + 118730557 T^{6} - 1546551482 T^{7} + 18786623971 T^{8} - 204028015834 T^{9} + 2076230010958 T^{10} - 19169270116548 T^{11} + 166309569944793 T^{12} - 1317191459607592 T^{13} + 166309569944793 p T^{14} - 19169270116548 p^{2} T^{15} + 2076230010958 p^{3} T^{16} - 204028015834 p^{4} T^{17} + 18786623971 p^{5} T^{18} - 1546551482 p^{6} T^{19} + 118730557 p^{7} T^{20} - 7904032 p^{8} T^{21} + 487840 p^{9} T^{22} - 24354 p^{10} T^{23} + 1126 p^{11} T^{24} - 34 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 - 24 T + 600 T^{2} - 8470 T^{3} + 127414 T^{4} - 1345560 T^{5} + 16160603 T^{6} - 146390276 T^{7} + 1587222753 T^{8} - 13132761180 T^{9} + 131651459570 T^{10} - 999411022758 T^{11} + 9266459176159 T^{12} - 65195774173224 T^{13} + 9266459176159 p T^{14} - 999411022758 p^{2} T^{15} + 131651459570 p^{3} T^{16} - 13132761180 p^{4} T^{17} + 1587222753 p^{5} T^{18} - 146390276 p^{6} T^{19} + 16160603 p^{7} T^{20} - 1345560 p^{8} T^{21} + 127414 p^{9} T^{22} - 8470 p^{10} T^{23} + 600 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 + 24 T + 639 T^{2} + 10644 T^{3} + 174946 T^{4} + 2328848 T^{5} + 29896130 T^{6} + 338557388 T^{7} + 3684265703 T^{8} + 36786820728 T^{9} + 354175444073 T^{10} + 3202181970944 T^{11} + 28024850856584 T^{12} + 233010451502400 T^{13} + 28024850856584 p T^{14} + 3202181970944 p^{2} T^{15} + 354175444073 p^{3} T^{16} + 36786820728 p^{4} T^{17} + 3684265703 p^{5} T^{18} + 338557388 p^{6} T^{19} + 29896130 p^{7} T^{20} + 2328848 p^{8} T^{21} + 174946 p^{9} T^{22} + 10644 p^{10} T^{23} + 639 p^{11} T^{24} + 24 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 - 20 T + 704 T^{2} - 10932 T^{3} + 226296 T^{4} - 2976698 T^{5} + 46454939 T^{6} - 536576738 T^{7} + 6886494809 T^{8} - 70839527368 T^{9} + 779972251182 T^{10} - 7187371362346 T^{11} + 69516223841919 T^{12} - 573542950848276 T^{13} + 69516223841919 p T^{14} - 7187371362346 p^{2} T^{15} + 779972251182 p^{3} T^{16} - 70839527368 p^{4} T^{17} + 6886494809 p^{5} T^{18} - 536576738 p^{6} T^{19} + 46454939 p^{7} T^{20} - 2976698 p^{8} T^{21} + 226296 p^{9} T^{22} - 10932 p^{10} T^{23} + 704 p^{11} T^{24} - 20 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 + 6 T + 493 T^{2} + 3000 T^{3} + 119078 T^{4} + 703636 T^{5} + 18959258 T^{6} + 104947624 T^{7} + 2256219623 T^{8} + 11490983050 T^{9} + 216572912651 T^{10} + 1019633689184 T^{11} + 17782423356976 T^{12} + 78706215594168 T^{13} + 17782423356976 p T^{14} + 1019633689184 p^{2} T^{15} + 216572912651 p^{3} T^{16} + 11490983050 p^{4} T^{17} + 2256219623 p^{5} T^{18} + 104947624 p^{6} T^{19} + 18959258 p^{7} T^{20} + 703636 p^{8} T^{21} + 119078 p^{9} T^{22} + 3000 p^{10} T^{23} + 493 p^{11} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 - 6 T + 692 T^{2} - 4728 T^{3} + 238171 T^{4} - 1709300 T^{5} + 53937609 T^{6} - 384030488 T^{7} + 8918377730 T^{8} - 60533914106 T^{9} + 1130331455095 T^{10} - 7103593813104 T^{11} + 112515318708438 T^{12} - 638491417420056 T^{13} + 112515318708438 p T^{14} - 7103593813104 p^{2} T^{15} + 1130331455095 p^{3} T^{16} - 60533914106 p^{4} T^{17} + 8918377730 p^{5} T^{18} - 384030488 p^{6} T^{19} + 53937609 p^{7} T^{20} - 1709300 p^{8} T^{21} + 238171 p^{9} T^{22} - 4728 p^{10} T^{23} + 692 p^{11} T^{24} - 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 - 14 T + 701 T^{2} - 9248 T^{3} + 243964 T^{4} - 3057612 T^{5} + 56112876 T^{6} - 660794592 T^{7} + 9490171973 T^{8} - 103503779058 T^{9} + 1241565929681 T^{10} - 12367680807936 T^{11} + 128734744958304 T^{12} - 1156076108853544 T^{13} + 128734744958304 p T^{14} - 12367680807936 p^{2} T^{15} + 1241565929681 p^{3} T^{16} - 103503779058 p^{4} T^{17} + 9490171973 p^{5} T^{18} - 660794592 p^{6} T^{19} + 56112876 p^{7} T^{20} - 3057612 p^{8} T^{21} + 243964 p^{9} T^{22} - 9248 p^{10} T^{23} + 701 p^{11} T^{24} - 14 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 36 T + 1015 T^{2} - 22220 T^{3} + 426252 T^{4} - 7209400 T^{5} + 111243952 T^{6} - 1574900804 T^{7} + 20757828361 T^{8} - 255085479596 T^{9} + 2948776460607 T^{10} - 32057113034800 T^{11} + 329367583670804 T^{12} - 3193780602283248 T^{13} + 329367583670804 p T^{14} - 32057113034800 p^{2} T^{15} + 2948776460607 p^{3} T^{16} - 255085479596 p^{4} T^{17} + 20757828361 p^{5} T^{18} - 1574900804 p^{6} T^{19} + 111243952 p^{7} T^{20} - 7209400 p^{8} T^{21} + 426252 p^{9} T^{22} - 22220 p^{10} T^{23} + 1015 p^{11} T^{24} - 36 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 + 32 T + 1262 T^{2} + 27996 T^{3} + 666729 T^{4} + 11673488 T^{5} + 211178773 T^{6} + 3093533548 T^{7} + 46149001680 T^{8} + 583363092192 T^{9} + 7464051220305 T^{10} + 82793760616248 T^{11} + 927248766522738 T^{12} + 9091625582573792 T^{13} + 927248766522738 p T^{14} + 82793760616248 p^{2} T^{15} + 7464051220305 p^{3} T^{16} + 583363092192 p^{4} T^{17} + 46149001680 p^{5} T^{18} + 3093533548 p^{6} T^{19} + 211178773 p^{7} T^{20} + 11673488 p^{8} T^{21} + 666729 p^{9} T^{22} + 27996 p^{10} T^{23} + 1262 p^{11} T^{24} + 32 p^{12} T^{25} + p^{13} T^{26} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.61465561781209859892577043745, −2.60026505834606179362179154168, −2.41089151413256864669082615307, −2.40293434680772583511384556568, −2.05548549383897235062313642530, −2.04936187224561601598931807629, −2.03787480968723545945471092555, −1.99251490051093599911200084910, −1.90418487573718907050059128251, −1.70038246915362805602152975267, −1.67649722249905047293742865406, −1.51810324450216146395525734421, −1.49458112756450719028670485718, −1.48548807760873715794242730428, −1.31699198787528858054837966950, −1.11584615998040081814307394552, −1.02617668033124755762251691852, −0.986776733679991985695914224488, −0.964295405503718346691954100226, −0.935460296389038525589942281066, −0.815011824107541595774030110046, −0.802911380197019686816249733735, −0.65354397691668194738343442919, −0.56062384928475488285171430315, −0.46847517899909028607948640072, 0.46847517899909028607948640072, 0.56062384928475488285171430315, 0.65354397691668194738343442919, 0.802911380197019686816249733735, 0.815011824107541595774030110046, 0.935460296389038525589942281066, 0.964295405503718346691954100226, 0.986776733679991985695914224488, 1.02617668033124755762251691852, 1.11584615998040081814307394552, 1.31699198787528858054837966950, 1.48548807760873715794242730428, 1.49458112756450719028670485718, 1.51810324450216146395525734421, 1.67649722249905047293742865406, 1.70038246915362805602152975267, 1.90418487573718907050059128251, 1.99251490051093599911200084910, 2.03787480968723545945471092555, 2.04936187224561601598931807629, 2.05548549383897235062313642530, 2.40293434680772583511384556568, 2.41089151413256864669082615307, 2.60026505834606179362179154168, 2.61465561781209859892577043745
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.