Dirichlet series
| L(s) = 1 | + 13·2-s − 13·3-s + 91·4-s + 3·5-s − 169·6-s + 5·7-s + 455·8-s + 91·9-s + 39·10-s + 8·11-s − 1.18e3·12-s + 13·13-s + 65·14-s − 39·15-s + 1.82e3·16-s − 17-s + 1.18e3·18-s + 5·19-s + 273·20-s − 65·21-s + 104·22-s − 9·23-s − 5.91e3·24-s − 16·25-s + 169·26-s − 455·27-s + 455·28-s + ⋯ |
| L(s) = 1 | + 9.19·2-s − 7.50·3-s + 91/2·4-s + 1.34·5-s − 68.9·6-s + 1.88·7-s + 160.·8-s + 91/3·9-s + 12.3·10-s + 2.41·11-s − 341.·12-s + 3.60·13-s + 17.3·14-s − 10.0·15-s + 455·16-s − 0.242·17-s + 278.·18-s + 1.14·19-s + 61.0·20-s − 14.1·21-s + 22.1·22-s − 1.87·23-s − 1.20e3·24-s − 3.19·25-s + 33.1·26-s − 87.5·27-s + 85.9·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{13} \cdot 3^{13} \cdot 13^{13} \cdot 103^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{13} \cdot 3^{13} \cdot 13^{13} \cdot 103^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(2^{13} \cdot 3^{13} \cdot 13^{13} \cdot 103^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.11685\times 10^{23}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 2^{13} \cdot 3^{13} \cdot 13^{13} \cdot 103^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(295735.7941\) |
| \(L(\frac12)\) | \(\approx\) | \(295735.7941\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{13} \) |
| 3 | \( ( 1 + T )^{13} \) | |
| 13 | \( ( 1 - T )^{13} \) | |
| 103 | \( ( 1 - T )^{13} \) | |
| good | 5 | \( 1 - 3 T + p^{2} T^{2} - 2 p^{2} T^{3} + 246 T^{4} - 199 T^{5} + 222 p T^{6} + 1404 T^{7} + 3074 T^{8} + 14037 T^{9} + 33619 T^{10} + 17011 T^{11} + 364541 T^{12} - 151604 T^{13} + 364541 p T^{14} + 17011 p^{2} T^{15} + 33619 p^{3} T^{16} + 14037 p^{4} T^{17} + 3074 p^{5} T^{18} + 1404 p^{6} T^{19} + 222 p^{8} T^{20} - 199 p^{8} T^{21} + 246 p^{9} T^{22} - 2 p^{12} T^{23} + p^{13} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) |
| 7 | \( 1 - 5 T + 66 T^{2} - 286 T^{3} + 299 p T^{4} - 7941 T^{5} + 42270 T^{6} - 20289 p T^{7} + 1780 p^{3} T^{8} - 1834001 T^{9} + 6718878 T^{10} - 18154644 T^{11} + 58451055 T^{12} - 142165308 T^{13} + 58451055 p T^{14} - 18154644 p^{2} T^{15} + 6718878 p^{3} T^{16} - 1834001 p^{4} T^{17} + 1780 p^{8} T^{18} - 20289 p^{7} T^{19} + 42270 p^{7} T^{20} - 7941 p^{8} T^{21} + 299 p^{10} T^{22} - 286 p^{10} T^{23} + 66 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 11 | \( 1 - 8 T + 81 T^{2} - 441 T^{3} + 2686 T^{4} - 10965 T^{5} + 4515 p T^{6} - 152162 T^{7} + 545672 T^{8} - 1092105 T^{9} + 3121168 T^{10} + 40003 T^{11} + 1547275 T^{12} + 66901436 T^{13} + 1547275 p T^{14} + 40003 p^{2} T^{15} + 3121168 p^{3} T^{16} - 1092105 p^{4} T^{17} + 545672 p^{5} T^{18} - 152162 p^{6} T^{19} + 4515 p^{8} T^{20} - 10965 p^{8} T^{21} + 2686 p^{9} T^{22} - 441 p^{10} T^{23} + 81 p^{11} T^{24} - 8 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 + T + 72 T^{2} + 122 T^{3} + 3269 T^{4} + 6415 T^{5} + 6467 p T^{6} + 226455 T^{7} + 2995534 T^{8} + 6146609 T^{9} + 68282485 T^{10} + 135831759 T^{11} + 1335831804 T^{12} + 2522899582 T^{13} + 1335831804 p T^{14} + 135831759 p^{2} T^{15} + 68282485 p^{3} T^{16} + 6146609 p^{4} T^{17} + 2995534 p^{5} T^{18} + 226455 p^{6} T^{19} + 6467 p^{8} T^{20} + 6415 p^{8} T^{21} + 3269 p^{9} T^{22} + 122 p^{10} T^{23} + 72 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 - 5 T + 115 T^{2} - 458 T^{3} + 6527 T^{4} - 21725 T^{5} + 250387 T^{6} - 709183 T^{7} + 7431436 T^{8} - 973125 p T^{9} + 185043908 T^{10} - 418680767 T^{11} + 4005392870 T^{12} - 8429455742 T^{13} + 4005392870 p T^{14} - 418680767 p^{2} T^{15} + 185043908 p^{3} T^{16} - 973125 p^{5} T^{17} + 7431436 p^{5} T^{18} - 709183 p^{6} T^{19} + 250387 p^{7} T^{20} - 21725 p^{8} T^{21} + 6527 p^{9} T^{22} - 458 p^{10} T^{23} + 115 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 + 9 T + 142 T^{2} + 928 T^{3} + 8808 T^{4} + 44074 T^{5} + 306041 T^{6} + 1052729 T^{7} + 5689972 T^{8} + 4307962 T^{9} + 11562851 T^{10} - 584517996 T^{11} - 2286303824 T^{12} - 20909499900 T^{13} - 2286303824 p T^{14} - 584517996 p^{2} T^{15} + 11562851 p^{3} T^{16} + 4307962 p^{4} T^{17} + 5689972 p^{5} T^{18} + 1052729 p^{6} T^{19} + 306041 p^{7} T^{20} + 44074 p^{8} T^{21} + 8808 p^{9} T^{22} + 928 p^{10} T^{23} + 142 p^{11} T^{24} + 9 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 + T + 166 T^{2} + 9 T^{3} + 13590 T^{4} - 9690 T^{5} + 742427 T^{6} - 954692 T^{7} + 30842049 T^{8} - 53084536 T^{9} + 1054198399 T^{10} - 2171709581 T^{11} + 32057467777 T^{12} - 70373835838 T^{13} + 32057467777 p T^{14} - 2171709581 p^{2} T^{15} + 1054198399 p^{3} T^{16} - 53084536 p^{4} T^{17} + 30842049 p^{5} T^{18} - 954692 p^{6} T^{19} + 742427 p^{7} T^{20} - 9690 p^{8} T^{21} + 13590 p^{9} T^{22} + 9 p^{10} T^{23} + 166 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 - 5 T + 271 T^{2} - 38 p T^{3} + 35903 T^{4} - 137885 T^{5} + 3096295 T^{6} - 10609183 T^{7} + 194619460 T^{8} - 598844151 T^{9} + 9436631708 T^{10} - 26157030311 T^{11} + 363528273854 T^{12} - 29250798770 p T^{13} + 363528273854 p T^{14} - 26157030311 p^{2} T^{15} + 9436631708 p^{3} T^{16} - 598844151 p^{4} T^{17} + 194619460 p^{5} T^{18} - 10609183 p^{6} T^{19} + 3096295 p^{7} T^{20} - 137885 p^{8} T^{21} + 35903 p^{9} T^{22} - 38 p^{11} T^{23} + 271 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 - 35 T + 879 T^{2} - 16087 T^{3} + 247562 T^{4} - 3231357 T^{5} + 37423176 T^{6} - 386418261 T^{7} + 3632397334 T^{8} - 31153758892 T^{9} + 246616407746 T^{10} - 1802194905228 T^{11} + 12241252447362 T^{12} - 77149226163688 T^{13} + 12241252447362 p T^{14} - 1802194905228 p^{2} T^{15} + 246616407746 p^{3} T^{16} - 31153758892 p^{4} T^{17} + 3632397334 p^{5} T^{18} - 386418261 p^{6} T^{19} + 37423176 p^{7} T^{20} - 3231357 p^{8} T^{21} + 247562 p^{9} T^{22} - 16087 p^{10} T^{23} + 879 p^{11} T^{24} - 35 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 24 T + 472 T^{2} - 7050 T^{3} + 94239 T^{4} - 1094474 T^{5} + 11715795 T^{6} - 113769865 T^{7} + 1033931287 T^{8} - 8696611730 T^{9} + 68942962671 T^{10} - 510748783110 T^{11} + 3581397425849 T^{12} - 23556186478802 T^{13} + 3581397425849 p T^{14} - 510748783110 p^{2} T^{15} + 68942962671 p^{3} T^{16} - 8696611730 p^{4} T^{17} + 1033931287 p^{5} T^{18} - 113769865 p^{6} T^{19} + 11715795 p^{7} T^{20} - 1094474 p^{8} T^{21} + 94239 p^{9} T^{22} - 7050 p^{10} T^{23} + 472 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 - 9 T + 381 T^{2} - 3018 T^{3} + 69158 T^{4} - 491152 T^{5} + 8049887 T^{6} - 51841958 T^{7} + 15802239 p T^{8} - 3991100381 T^{9} + 44419157527 T^{10} - 238235230232 T^{11} + 2334269136875 T^{12} - 11387991993860 T^{13} + 2334269136875 p T^{14} - 238235230232 p^{2} T^{15} + 44419157527 p^{3} T^{16} - 3991100381 p^{4} T^{17} + 15802239 p^{6} T^{18} - 51841958 p^{6} T^{19} + 8049887 p^{7} T^{20} - 491152 p^{8} T^{21} + 69158 p^{9} T^{22} - 3018 p^{10} T^{23} + 381 p^{11} T^{24} - 9 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 + 8 T + 375 T^{2} + 2199 T^{3} + 62628 T^{4} + 257805 T^{5} + 6346256 T^{6} + 15847093 T^{7} + 448993624 T^{8} + 394121817 T^{9} + 24578534949 T^{10} - 14898890565 T^{11} + 1174103421532 T^{12} - 1530957472620 T^{13} + 1174103421532 p T^{14} - 14898890565 p^{2} T^{15} + 24578534949 p^{3} T^{16} + 394121817 p^{4} T^{17} + 448993624 p^{5} T^{18} + 15847093 p^{6} T^{19} + 6346256 p^{7} T^{20} + 257805 p^{8} T^{21} + 62628 p^{9} T^{22} + 2199 p^{10} T^{23} + 375 p^{11} T^{24} + 8 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 - 32 T + 703 T^{2} - 11193 T^{3} + 150665 T^{4} - 1696185 T^{5} + 16968079 T^{6} - 148721494 T^{7} + 1192804159 T^{8} - 8603867786 T^{9} + 58893158913 T^{10} - 379379490929 T^{11} + 2523907028612 T^{12} - 17333659447546 T^{13} + 2523907028612 p T^{14} - 379379490929 p^{2} T^{15} + 58893158913 p^{3} T^{16} - 8603867786 p^{4} T^{17} + 1192804159 p^{5} T^{18} - 148721494 p^{6} T^{19} + 16968079 p^{7} T^{20} - 1696185 p^{8} T^{21} + 150665 p^{9} T^{22} - 11193 p^{10} T^{23} + 703 p^{11} T^{24} - 32 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 - 12 T + 492 T^{2} - 5819 T^{3} + 123502 T^{4} - 1361556 T^{5} + 20687717 T^{6} - 206164863 T^{7} + 2540047898 T^{8} - 22702133281 T^{9} + 239075358515 T^{10} - 1918882189879 T^{11} + 17700735919831 T^{12} - 127370740797940 T^{13} + 17700735919831 p T^{14} - 1918882189879 p^{2} T^{15} + 239075358515 p^{3} T^{16} - 22702133281 p^{4} T^{17} + 2540047898 p^{5} T^{18} - 206164863 p^{6} T^{19} + 20687717 p^{7} T^{20} - 1361556 p^{8} T^{21} + 123502 p^{9} T^{22} - 5819 p^{10} T^{23} + 492 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 - 17 T + 395 T^{2} - 5414 T^{3} + 81146 T^{4} - 901986 T^{5} + 10902127 T^{6} - 106300462 T^{7} + 1104348671 T^{8} - 9720783669 T^{9} + 91269537043 T^{10} - 736154617740 T^{11} + 6374523735013 T^{12} - 48129864687696 T^{13} + 6374523735013 p T^{14} - 736154617740 p^{2} T^{15} + 91269537043 p^{3} T^{16} - 9720783669 p^{4} T^{17} + 1104348671 p^{5} T^{18} - 106300462 p^{6} T^{19} + 10902127 p^{7} T^{20} - 901986 p^{8} T^{21} + 81146 p^{9} T^{22} - 5414 p^{10} T^{23} + 395 p^{11} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 - 20 T + 601 T^{2} - 9365 T^{3} + 164472 T^{4} - 2126310 T^{5} + 28028480 T^{6} - 311675821 T^{7} + 3396003920 T^{8} - 33461413283 T^{9} + 318622545371 T^{10} - 2859111247421 T^{11} + 24737883791037 T^{12} - 206153123607736 T^{13} + 24737883791037 p T^{14} - 2859111247421 p^{2} T^{15} + 318622545371 p^{3} T^{16} - 33461413283 p^{4} T^{17} + 3396003920 p^{5} T^{18} - 311675821 p^{6} T^{19} + 28028480 p^{7} T^{20} - 2126310 p^{8} T^{21} + 164472 p^{9} T^{22} - 9365 p^{10} T^{23} + 601 p^{11} T^{24} - 20 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 - 16 T + 650 T^{2} - 8168 T^{3} + 187959 T^{4} - 1957904 T^{5} + 33394406 T^{6} - 299247255 T^{7} + 4217856563 T^{8} - 33459809651 T^{9} + 414713355965 T^{10} - 2988823921021 T^{11} + 33936031084590 T^{12} - 227145830014434 T^{13} + 33936031084590 p T^{14} - 2988823921021 p^{2} T^{15} + 414713355965 p^{3} T^{16} - 33459809651 p^{4} T^{17} + 4217856563 p^{5} T^{18} - 299247255 p^{6} T^{19} + 33394406 p^{7} T^{20} - 1957904 p^{8} T^{21} + 187959 p^{9} T^{22} - 8168 p^{10} T^{23} + 650 p^{11} T^{24} - 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 46 T + 1720 T^{2} - 44305 T^{3} + 993024 T^{4} - 18358223 T^{5} + 305680033 T^{6} - 4459620559 T^{7} + 59766164049 T^{8} - 721821551394 T^{9} + 8101777460075 T^{10} - 83131105504269 T^{11} + 797678453714516 T^{12} - 7040793819274228 T^{13} + 797678453714516 p T^{14} - 83131105504269 p^{2} T^{15} + 8101777460075 p^{3} T^{16} - 721821551394 p^{4} T^{17} + 59766164049 p^{5} T^{18} - 4459620559 p^{6} T^{19} + 305680033 p^{7} T^{20} - 18358223 p^{8} T^{21} + 993024 p^{9} T^{22} - 44305 p^{10} T^{23} + 1720 p^{11} T^{24} - 46 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 - 17 T + 6 p T^{2} - 5754 T^{3} + 109720 T^{4} - 1110821 T^{5} + 17405351 T^{6} - 152496831 T^{7} + 2129325399 T^{8} - 16573632681 T^{9} + 215111030146 T^{10} - 1525675616183 T^{11} + 18938845788835 T^{12} - 126205446995874 T^{13} + 18938845788835 p T^{14} - 1525675616183 p^{2} T^{15} + 215111030146 p^{3} T^{16} - 16573632681 p^{4} T^{17} + 2129325399 p^{5} T^{18} - 152496831 p^{6} T^{19} + 17405351 p^{7} T^{20} - 1110821 p^{8} T^{21} + 109720 p^{9} T^{22} - 5754 p^{10} T^{23} + 6 p^{12} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 - 13 T + 525 T^{2} - 5210 T^{3} + 124484 T^{4} - 1002252 T^{5} + 19074120 T^{6} - 130370586 T^{7} + 2254421737 T^{8} - 13369356510 T^{9} + 224626910605 T^{10} - 1169892326755 T^{11} + 19915374871540 T^{12} - 96528686048908 T^{13} + 19915374871540 p T^{14} - 1169892326755 p^{2} T^{15} + 224626910605 p^{3} T^{16} - 13369356510 p^{4} T^{17} + 2254421737 p^{5} T^{18} - 130370586 p^{6} T^{19} + 19074120 p^{7} T^{20} - 1002252 p^{8} T^{21} + 124484 p^{9} T^{22} - 5210 p^{10} T^{23} + 525 p^{11} T^{24} - 13 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 14 T + 893 T^{2} - 9976 T^{3} + 358111 T^{4} - 3212529 T^{5} + 86874731 T^{6} - 623018848 T^{7} + 14557512254 T^{8} - 83111773324 T^{9} + 1844464004430 T^{10} - 8575604633208 T^{11} + 190802854208908 T^{12} - 782594002264202 T^{13} + 190802854208908 p T^{14} - 8575604633208 p^{2} T^{15} + 1844464004430 p^{3} T^{16} - 83111773324 p^{4} T^{17} + 14557512254 p^{5} T^{18} - 623018848 p^{6} T^{19} + 86874731 p^{7} T^{20} - 3212529 p^{8} T^{21} + 358111 p^{9} T^{22} - 9976 p^{10} T^{23} + 893 p^{11} T^{24} - 14 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 - 46 T + 1814 T^{2} - 48110 T^{3} + 1136778 T^{4} - 21857435 T^{5} + 384959219 T^{6} - 5892700340 T^{7} + 84305542616 T^{8} - 1084305973884 T^{9} + 13249273926983 T^{10} - 148620823050822 T^{11} + 1603545598731357 T^{12} - 16063777032627910 T^{13} + 1603545598731357 p T^{14} - 148620823050822 p^{2} T^{15} + 13249273926983 p^{3} T^{16} - 1084305973884 p^{4} T^{17} + 84305542616 p^{5} T^{18} - 5892700340 p^{6} T^{19} + 384959219 p^{7} T^{20} - 21857435 p^{8} T^{21} + 1136778 p^{9} T^{22} - 48110 p^{10} T^{23} + 1814 p^{11} T^{24} - 46 p^{12} T^{25} + p^{13} T^{26} \) | |
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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
−1.90854106772840480328941850875, −1.88669017825097704106811181395, −1.88083419318736556095840417263, −1.87934732716211931515496881626, −1.85274557138056380200779833783, −1.81010136810805156098473998929, −1.78521768121254993712470995364, −1.75634363217194628710877831850, −1.74960552904095600161094708289, −1.73289100135665329720605039400, −1.72411060576112674482187778007, −1.61635363864469993542225987455, −1.14028872540681543323075013889, −1.10168956351415920156086589568, −1.07550989015818563090429926063, −0.985194474899459416665610195015, −0.941807004700099105778936571801, −0.819378235836736284037420836305, −0.75480199887115747638491945811, −0.71533360484807405265475362636, −0.71435626186233512978061217317, −0.65262893997585866203527531551, −0.54450959474699517871506938521, −0.48578500815436222927709447469, −0.45880289925356086404265917917, 0.45880289925356086404265917917, 0.48578500815436222927709447469, 0.54450959474699517871506938521, 0.65262893997585866203527531551, 0.71435626186233512978061217317, 0.71533360484807405265475362636, 0.75480199887115747638491945811, 0.819378235836736284037420836305, 0.941807004700099105778936571801, 0.985194474899459416665610195015, 1.07550989015818563090429926063, 1.10168956351415920156086589568, 1.14028872540681543323075013889, 1.61635363864469993542225987455, 1.72411060576112674482187778007, 1.73289100135665329720605039400, 1.74960552904095600161094708289, 1.75634363217194628710877831850, 1.78521768121254993712470995364, 1.81010136810805156098473998929, 1.85274557138056380200779833783, 1.87934732716211931515496881626, 1.88083419318736556095840417263, 1.88669017825097704106811181395, 1.90854106772840480328941850875
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.