Dirichlet series
| L(s) = 1 | + 4·3-s + 5·5-s − 6·7-s − 9-s − 6·11-s − 13·13-s + 20·15-s + 4·17-s + 3·19-s − 24·21-s − 13·23-s − 8·25-s − 24·27-s + 13·29-s − 21·31-s − 24·33-s − 30·35-s + 23·37-s − 52·39-s + 4·41-s + 24·43-s − 5·45-s − 7·47-s − 7·49-s + 16·51-s + 17·53-s − 30·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2.23·5-s − 2.26·7-s − 1/3·9-s − 1.80·11-s − 3.60·13-s + 5.16·15-s + 0.970·17-s + 0.688·19-s − 5.23·21-s − 2.71·23-s − 8/5·25-s − 4.61·27-s + 2.41·29-s − 3.77·31-s − 4.17·33-s − 5.07·35-s + 3.78·37-s − 8.32·39-s + 0.624·41-s + 3.65·43-s − 0.745·45-s − 1.02·47-s − 49-s + 2.24·51-s + 2.33·53-s − 4.04·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{52} \cdot 13^{13} \cdot 29^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{52} \cdot 13^{13} \cdot 29^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(2^{52} \cdot 13^{13} \cdot 29^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.50930\times 10^{21}\) |
| Root analytic conductor: | \(6.94015\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 2^{52} \cdot 13^{13} \cdot 29^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.52628771\) |
| \(L(\frac12)\) | \(\approx\) | \(10.52628771\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 + T )^{13} \) | |
| 29 | \( ( 1 - T )^{13} \) | |
| good | 3 | \( 1 - 4 T + 17 T^{2} - 16 p T^{3} + 5 p^{3} T^{4} - 323 T^{5} + 763 T^{6} - 1699 T^{7} + 3677 T^{8} - 7538 T^{9} + 14953 T^{10} - 27937 T^{11} + 51244 T^{12} - 88838 T^{13} + 51244 p T^{14} - 27937 p^{2} T^{15} + 14953 p^{3} T^{16} - 7538 p^{4} T^{17} + 3677 p^{5} T^{18} - 1699 p^{6} T^{19} + 763 p^{7} T^{20} - 323 p^{8} T^{21} + 5 p^{12} T^{22} - 16 p^{11} T^{23} + 17 p^{11} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) |
| 5 | \( 1 - p T + 33 T^{2} - 128 T^{3} + 114 p T^{4} - 1899 T^{5} + 6622 T^{6} - 19491 T^{7} + 58126 T^{8} - 155182 T^{9} + 409254 T^{10} - 1002337 T^{11} + 481038 p T^{12} - 1089868 p T^{13} + 481038 p^{2} T^{14} - 1002337 p^{2} T^{15} + 409254 p^{3} T^{16} - 155182 p^{4} T^{17} + 58126 p^{5} T^{18} - 19491 p^{6} T^{19} + 6622 p^{7} T^{20} - 1899 p^{8} T^{21} + 114 p^{10} T^{22} - 128 p^{10} T^{23} + 33 p^{11} T^{24} - p^{13} T^{25} + p^{13} T^{26} \) | |
| 7 | \( 1 + 6 T + 43 T^{2} + 180 T^{3} + 887 T^{4} + 3232 T^{5} + 13161 T^{6} + 42389 T^{7} + 149827 T^{8} + 439519 T^{9} + 1411009 T^{10} + 3832523 T^{11} + 11340466 T^{12} + 28705002 T^{13} + 11340466 p T^{14} + 3832523 p^{2} T^{15} + 1411009 p^{3} T^{16} + 439519 p^{4} T^{17} + 149827 p^{5} T^{18} + 42389 p^{6} T^{19} + 13161 p^{7} T^{20} + 3232 p^{8} T^{21} + 887 p^{9} T^{22} + 180 p^{10} T^{23} + 43 p^{11} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 11 | \( 1 + 6 T + 71 T^{2} + 394 T^{3} + 2536 T^{4} + 12797 T^{5} + 63058 T^{6} + 280229 T^{7} + 1204451 T^{8} + 4732226 T^{9} + 18393185 T^{10} + 66249585 T^{11} + 234399910 T^{12} + 71744658 p T^{13} + 234399910 p T^{14} + 66249585 p^{2} T^{15} + 18393185 p^{3} T^{16} + 4732226 p^{4} T^{17} + 1204451 p^{5} T^{18} + 280229 p^{6} T^{19} + 63058 p^{7} T^{20} + 12797 p^{8} T^{21} + 2536 p^{9} T^{22} + 394 p^{10} T^{23} + 71 p^{11} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 - 4 T + 114 T^{2} - 411 T^{3} + 6343 T^{4} - 21654 T^{5} + 13875 p T^{6} - 46664 p T^{7} + 6690126 T^{8} - 22449958 T^{9} + 154623179 T^{10} - 510657677 T^{11} + 3029016842 T^{12} - 9538149280 T^{13} + 3029016842 p T^{14} - 510657677 p^{2} T^{15} + 154623179 p^{3} T^{16} - 22449958 p^{4} T^{17} + 6690126 p^{5} T^{18} - 46664 p^{7} T^{19} + 13875 p^{8} T^{20} - 21654 p^{8} T^{21} + 6343 p^{9} T^{22} - 411 p^{10} T^{23} + 114 p^{11} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 - 3 T + 119 T^{2} - 488 T^{3} + 7827 T^{4} - 35081 T^{5} + 369025 T^{6} - 1632793 T^{7} + 13314485 T^{8} - 56181349 T^{9} + 381258971 T^{10} - 1499806783 T^{11} + 8861716182 T^{12} - 31819437102 T^{13} + 8861716182 p T^{14} - 1499806783 p^{2} T^{15} + 381258971 p^{3} T^{16} - 56181349 p^{4} T^{17} + 13314485 p^{5} T^{18} - 1632793 p^{6} T^{19} + 369025 p^{7} T^{20} - 35081 p^{8} T^{21} + 7827 p^{9} T^{22} - 488 p^{10} T^{23} + 119 p^{11} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 + 13 T + 186 T^{2} + 1475 T^{3} + 13095 T^{4} + 85703 T^{5} + 639181 T^{6} + 3857805 T^{7} + 24959700 T^{8} + 136521059 T^{9} + 781825403 T^{10} + 3995204992 T^{11} + 21119456214 T^{12} + 100589444242 T^{13} + 21119456214 p T^{14} + 3995204992 p^{2} T^{15} + 781825403 p^{3} T^{16} + 136521059 p^{4} T^{17} + 24959700 p^{5} T^{18} + 3857805 p^{6} T^{19} + 639181 p^{7} T^{20} + 85703 p^{8} T^{21} + 13095 p^{9} T^{22} + 1475 p^{10} T^{23} + 186 p^{11} T^{24} + 13 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 + 21 T + 338 T^{2} + 3717 T^{3} + 36393 T^{4} + 297763 T^{5} + 2346555 T^{6} + 16848845 T^{7} + 120184496 T^{8} + 795883695 T^{9} + 5174599983 T^{10} + 31232373614 T^{11} + 185385469410 T^{12} + 1036095570482 T^{13} + 185385469410 p T^{14} + 31232373614 p^{2} T^{15} + 5174599983 p^{3} T^{16} + 795883695 p^{4} T^{17} + 120184496 p^{5} T^{18} + 16848845 p^{6} T^{19} + 2346555 p^{7} T^{20} + 297763 p^{8} T^{21} + 36393 p^{9} T^{22} + 3717 p^{10} T^{23} + 338 p^{11} T^{24} + 21 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 - 23 T + 384 T^{2} - 4600 T^{3} + 45694 T^{4} - 378284 T^{5} + 2796253 T^{6} - 18483601 T^{7} + 116761839 T^{8} - 705623147 T^{9} + 4297586092 T^{10} - 25850329807 T^{11} + 158655266909 T^{12} - 955579188932 T^{13} + 158655266909 p T^{14} - 25850329807 p^{2} T^{15} + 4297586092 p^{3} T^{16} - 705623147 p^{4} T^{17} + 116761839 p^{5} T^{18} - 18483601 p^{6} T^{19} + 2796253 p^{7} T^{20} - 378284 p^{8} T^{21} + 45694 p^{9} T^{22} - 4600 p^{10} T^{23} + 384 p^{11} T^{24} - 23 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 4 T + 302 T^{2} - 980 T^{3} + 45239 T^{4} - 120470 T^{5} + 4479469 T^{6} - 9656413 T^{7} + 329223788 T^{8} - 569416688 T^{9} + 19211458473 T^{10} - 27253929191 T^{11} + 929090675736 T^{12} - 1159542133788 T^{13} + 929090675736 p T^{14} - 27253929191 p^{2} T^{15} + 19211458473 p^{3} T^{16} - 569416688 p^{4} T^{17} + 329223788 p^{5} T^{18} - 9656413 p^{6} T^{19} + 4479469 p^{7} T^{20} - 120470 p^{8} T^{21} + 45239 p^{9} T^{22} - 980 p^{10} T^{23} + 302 p^{11} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 - 24 T + 457 T^{2} - 5852 T^{3} + 63861 T^{4} - 544565 T^{5} + 4092957 T^{6} - 24770617 T^{7} + 139886311 T^{8} - 672315918 T^{9} + 3945036159 T^{10} - 23641016903 T^{11} + 185581623284 T^{12} - 1178934252226 T^{13} + 185581623284 p T^{14} - 23641016903 p^{2} T^{15} + 3945036159 p^{3} T^{16} - 672315918 p^{4} T^{17} + 139886311 p^{5} T^{18} - 24770617 p^{6} T^{19} + 4092957 p^{7} T^{20} - 544565 p^{8} T^{21} + 63861 p^{9} T^{22} - 5852 p^{10} T^{23} + 457 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 + 7 T + 313 T^{2} + 1736 T^{3} + 46319 T^{4} + 194906 T^{5} + 4350473 T^{6} + 12915955 T^{7} + 297552615 T^{8} + 538861939 T^{9} + 16388462067 T^{10} + 14153085301 T^{11} + 801511132110 T^{12} + 374983800184 T^{13} + 801511132110 p T^{14} + 14153085301 p^{2} T^{15} + 16388462067 p^{3} T^{16} + 538861939 p^{4} T^{17} + 297552615 p^{5} T^{18} + 12915955 p^{6} T^{19} + 4350473 p^{7} T^{20} + 194906 p^{8} T^{21} + 46319 p^{9} T^{22} + 1736 p^{10} T^{23} + 313 p^{11} T^{24} + 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 - 17 T + 491 T^{2} - 6377 T^{3} + 106907 T^{4} - 1143115 T^{5} + 14475141 T^{6} - 134409273 T^{7} + 1428423931 T^{8} - 11927680547 T^{9} + 111843899417 T^{10} - 16067764574 p T^{11} + 7184577872036 T^{12} - 49820515168706 T^{13} + 7184577872036 p T^{14} - 16067764574 p^{3} T^{15} + 111843899417 p^{3} T^{16} - 11927680547 p^{4} T^{17} + 1428423931 p^{5} T^{18} - 134409273 p^{6} T^{19} + 14475141 p^{7} T^{20} - 1143115 p^{8} T^{21} + 106907 p^{9} T^{22} - 6377 p^{10} T^{23} + 491 p^{11} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 + 11 T + 329 T^{2} + 2781 T^{3} + 52785 T^{4} + 370309 T^{5} + 5761539 T^{6} + 34884467 T^{7} + 491273273 T^{8} + 2608348945 T^{9} + 35067650697 T^{10} + 166344337456 T^{11} + 2220587244262 T^{12} + 9909486758574 T^{13} + 2220587244262 p T^{14} + 166344337456 p^{2} T^{15} + 35067650697 p^{3} T^{16} + 2608348945 p^{4} T^{17} + 491273273 p^{5} T^{18} + 34884467 p^{6} T^{19} + 5761539 p^{7} T^{20} + 370309 p^{8} T^{21} + 52785 p^{9} T^{22} + 2781 p^{10} T^{23} + 329 p^{11} T^{24} + 11 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 - 26 T + 592 T^{2} - 9096 T^{3} + 129763 T^{4} - 1488528 T^{5} + 16258273 T^{6} - 150375651 T^{7} + 1352540642 T^{8} - 10511435850 T^{9} + 82656689749 T^{10} - 575618965685 T^{11} + 4415745087244 T^{12} - 31699471236712 T^{13} + 4415745087244 p T^{14} - 575618965685 p^{2} T^{15} + 82656689749 p^{3} T^{16} - 10511435850 p^{4} T^{17} + 1352540642 p^{5} T^{18} - 150375651 p^{6} T^{19} + 16258273 p^{7} T^{20} - 1488528 p^{8} T^{21} + 129763 p^{9} T^{22} - 9096 p^{10} T^{23} + 592 p^{11} T^{24} - 26 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 - 35 T + 1048 T^{2} - 20839 T^{3} + 364003 T^{4} - 5020725 T^{5} + 61572523 T^{6} - 614967331 T^{7} + 5373327568 T^{8} - 35604283125 T^{9} + 175920176197 T^{10} - 60804084990 T^{11} - 7239824924828 T^{12} + 93996606838778 T^{13} - 7239824924828 p T^{14} - 60804084990 p^{2} T^{15} + 175920176197 p^{3} T^{16} - 35604283125 p^{4} T^{17} + 5373327568 p^{5} T^{18} - 614967331 p^{6} T^{19} + 61572523 p^{7} T^{20} - 5020725 p^{8} T^{21} + 364003 p^{9} T^{22} - 20839 p^{10} T^{23} + 1048 p^{11} T^{24} - 35 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 + 30 T + 775 T^{2} + 13788 T^{3} + 215859 T^{4} + 2808789 T^{5} + 33651909 T^{6} + 362052255 T^{7} + 3762137743 T^{8} + 36922051692 T^{9} + 359216525593 T^{10} + 3334502374465 T^{11} + 30346951575722 T^{12} + 259731390757866 T^{13} + 30346951575722 p T^{14} + 3334502374465 p^{2} T^{15} + 359216525593 p^{3} T^{16} + 36922051692 p^{4} T^{17} + 3762137743 p^{5} T^{18} + 362052255 p^{6} T^{19} + 33651909 p^{7} T^{20} + 2808789 p^{8} T^{21} + 215859 p^{9} T^{22} + 13788 p^{10} T^{23} + 775 p^{11} T^{24} + 30 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 17 T + 494 T^{2} - 6749 T^{3} + 116582 T^{4} - 1324812 T^{5} + 18079037 T^{6} - 180468300 T^{7} + 2159948997 T^{8} - 19850592055 T^{9} + 216663343472 T^{10} - 1856961240351 T^{11} + 18634791500433 T^{12} - 147550779385320 T^{13} + 18634791500433 p T^{14} - 1856961240351 p^{2} T^{15} + 216663343472 p^{3} T^{16} - 19850592055 p^{4} T^{17} + 2159948997 p^{5} T^{18} - 180468300 p^{6} T^{19} + 18079037 p^{7} T^{20} - 1324812 p^{8} T^{21} + 116582 p^{9} T^{22} - 6749 p^{10} T^{23} + 494 p^{11} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 + 3 T + 527 T^{2} + 960 T^{3} + 137573 T^{4} + 158091 T^{5} + 23872913 T^{6} + 19851327 T^{7} + 3114683991 T^{8} + 2332865437 T^{9} + 328687206197 T^{10} + 249892115757 T^{11} + 29548608752496 T^{12} + 22140087267690 T^{13} + 29548608752496 p T^{14} + 249892115757 p^{2} T^{15} + 328687206197 p^{3} T^{16} + 2332865437 p^{4} T^{17} + 3114683991 p^{5} T^{18} + 19851327 p^{6} T^{19} + 23872913 p^{7} T^{20} + 158091 p^{8} T^{21} + 137573 p^{9} T^{22} + 960 p^{10} T^{23} + 527 p^{11} T^{24} + 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 - 18 T + 704 T^{2} - 10855 T^{3} + 237392 T^{4} - 3198091 T^{5} + 51420570 T^{6} - 614644897 T^{7} + 8110358636 T^{8} - 87055579922 T^{9} + 998918556545 T^{10} - 9717871946664 T^{11} + 100045337976042 T^{12} - 886801786453842 T^{13} + 100045337976042 p T^{14} - 9717871946664 p^{2} T^{15} + 998918556545 p^{3} T^{16} - 87055579922 p^{4} T^{17} + 8110358636 p^{5} T^{18} - 614644897 p^{6} T^{19} + 51420570 p^{7} T^{20} - 3198091 p^{8} T^{21} + 237392 p^{9} T^{22} - 10855 p^{10} T^{23} + 704 p^{11} T^{24} - 18 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 38 T + 1337 T^{2} - 30723 T^{3} + 648463 T^{4} - 11005026 T^{5} + 174114777 T^{6} - 2385832714 T^{7} + 30976344501 T^{8} - 362514590824 T^{9} + 4083986682841 T^{10} - 42552749377347 T^{11} + 432316708982040 T^{12} - 4119697464478784 T^{13} + 432316708982040 p T^{14} - 42552749377347 p^{2} T^{15} + 4083986682841 p^{3} T^{16} - 362514590824 p^{4} T^{17} + 30976344501 p^{5} T^{18} - 2385832714 p^{6} T^{19} + 174114777 p^{7} T^{20} - 11005026 p^{8} T^{21} + 648463 p^{9} T^{22} - 30723 p^{10} T^{23} + 1337 p^{11} T^{24} - 38 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 - 7 T + 797 T^{2} - 5710 T^{3} + 300291 T^{4} - 2195391 T^{5} + 71141259 T^{6} - 533060077 T^{7} + 12001655147 T^{8} - 92635394083 T^{9} + 1571519499607 T^{10} - 12388341482261 T^{11} + 172393869127218 T^{12} - 1330834194630926 T^{13} + 172393869127218 p T^{14} - 12388341482261 p^{2} T^{15} + 1571519499607 p^{3} T^{16} - 92635394083 p^{4} T^{17} + 12001655147 p^{5} T^{18} - 533060077 p^{6} T^{19} + 71141259 p^{7} T^{20} - 2195391 p^{8} T^{21} + 300291 p^{9} T^{22} - 5710 p^{10} T^{23} + 797 p^{11} T^{24} - 7 p^{12} T^{25} + p^{13} T^{26} \) | |
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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
−2.26790298696080672658101129478, −2.18906957593139910873001017666, −2.09410217116194855543591038498, −2.05588633466572585206506097438, −2.01501598568013938055464698651, −2.00627000024304855397284778389, −1.92851167282776198195295058468, −1.84763430556259651533645189714, −1.79442540453294155312174259569, −1.72504751689071573878628153893, −1.57952339424998579736439862456, −1.56463290633012434312904734782, −1.37665781114158077325794606502, −1.19411563934606120978688305648, −1.11921011584976000299261301798, −0.849660778802930960401262892447, −0.790893099276880608581936049301, −0.76986933175189689477775330681, −0.62963711441086371689216832334, −0.58543070792107086816537992505, −0.53872120728769774675575899162, −0.44588911288845435764734570488, −0.40199817155328915146160855887, −0.32734449028126003799998692174, −0.06941363171385168480022107034, 0.06941363171385168480022107034, 0.32734449028126003799998692174, 0.40199817155328915146160855887, 0.44588911288845435764734570488, 0.53872120728769774675575899162, 0.58543070792107086816537992505, 0.62963711441086371689216832334, 0.76986933175189689477775330681, 0.790893099276880608581936049301, 0.849660778802930960401262892447, 1.11921011584976000299261301798, 1.19411563934606120978688305648, 1.37665781114158077325794606502, 1.56463290633012434312904734782, 1.57952339424998579736439862456, 1.72504751689071573878628153893, 1.79442540453294155312174259569, 1.84763430556259651533645189714, 1.92851167282776198195295058468, 2.00627000024304855397284778389, 2.01501598568013938055464698651, 2.05588633466572585206506097438, 2.09410217116194855543591038498, 2.18906957593139910873001017666, 2.26790298696080672658101129478
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.