Dirichlet series
| L(s) = 1 | + 13·2-s + 13·3-s + 91·4-s + 3·5-s + 169·6-s + 12·7-s + 455·8-s + 91·9-s + 39·10-s + 4·11-s + 1.18e3·12-s + 18·13-s + 156·14-s + 39·15-s + 1.82e3·16-s + 3·17-s + 1.18e3·18-s + 13·19-s + 273·20-s + 156·21-s + 52·22-s + 8·23-s + 5.91e3·24-s − 16·25-s + 234·26-s + 455·27-s + 1.09e3·28-s + ⋯ |
| L(s) = 1 | + 9.19·2-s + 7.50·3-s + 91/2·4-s + 1.34·5-s + 68.9·6-s + 4.53·7-s + 160.·8-s + 91/3·9-s + 12.3·10-s + 1.20·11-s + 341.·12-s + 4.99·13-s + 41.6·14-s + 10.0·15-s + 455·16-s + 0.727·17-s + 278.·18-s + 2.98·19-s + 61.0·20-s + 34.0·21-s + 11.0·22-s + 1.66·23-s + 1.20e3·24-s − 3.19·25-s + 45.8·26-s + 87.5·27-s + 206.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{13} \cdot 3^{13} \cdot 19^{13} \cdot 53^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{13} \cdot 3^{13} \cdot 19^{13} \cdot 53^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(2^{13} \cdot 3^{13} \cdot 19^{13} \cdot 53^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.67276\times 10^{21}\) |
| Root analytic conductor: | \(6.94590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 2^{13} \cdot 3^{13} \cdot 19^{13} \cdot 53^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.898985563\times10^{8}\) |
| \(L(\frac12)\) | \(\approx\) | \(2.898985563\times10^{8}\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{13} \) | |
| 19 | \( ( 1 - T )^{13} \) | |
| 53 | \( ( 1 + T )^{13} \) | |
| good | 5 | \( 1 - 3 T + p^{2} T^{2} - 57 T^{3} + 287 T^{4} - 507 T^{5} + 2001 T^{6} - 2766 T^{7} + 10123 T^{8} - 2211 p T^{9} + 44509 T^{10} - 42813 T^{11} + 204314 T^{12} - 38886 p T^{13} + 204314 p T^{14} - 42813 p^{2} T^{15} + 44509 p^{3} T^{16} - 2211 p^{5} T^{17} + 10123 p^{5} T^{18} - 2766 p^{6} T^{19} + 2001 p^{7} T^{20} - 507 p^{8} T^{21} + 287 p^{9} T^{22} - 57 p^{10} T^{23} + p^{13} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) |
| 7 | \( 1 - 12 T + 97 T^{2} - 562 T^{3} + 2650 T^{4} - 10334 T^{5} + 4990 p T^{6} - 104063 T^{7} + 288341 T^{8} - 774238 T^{9} + 2144461 T^{10} - 6089343 T^{11} + 17381680 T^{12} - 47202352 T^{13} + 17381680 p T^{14} - 6089343 p^{2} T^{15} + 2144461 p^{3} T^{16} - 774238 p^{4} T^{17} + 288341 p^{5} T^{18} - 104063 p^{6} T^{19} + 4990 p^{8} T^{20} - 10334 p^{8} T^{21} + 2650 p^{9} T^{22} - 562 p^{10} T^{23} + 97 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 11 | \( 1 - 4 T + 68 T^{2} - 163 T^{3} + 2044 T^{4} - 2586 T^{5} + 40824 T^{6} - 16517 T^{7} + 662738 T^{8} + 127192 T^{9} + 9314419 T^{10} + 5165200 T^{11} + 114475028 T^{12} + 76332556 T^{13} + 114475028 p T^{14} + 5165200 p^{2} T^{15} + 9314419 p^{3} T^{16} + 127192 p^{4} T^{17} + 662738 p^{5} T^{18} - 16517 p^{6} T^{19} + 40824 p^{7} T^{20} - 2586 p^{8} T^{21} + 2044 p^{9} T^{22} - 163 p^{10} T^{23} + 68 p^{11} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 13 | \( 1 - 18 T + 229 T^{2} - 2230 T^{3} + 18426 T^{4} - 132180 T^{5} + 849664 T^{6} - 4944275 T^{7} + 26403235 T^{8} - 130094222 T^{9} + 595492543 T^{10} - 2538749167 T^{11} + 10115848214 T^{12} - 37692765128 T^{13} + 10115848214 p T^{14} - 2538749167 p^{2} T^{15} + 595492543 p^{3} T^{16} - 130094222 p^{4} T^{17} + 26403235 p^{5} T^{18} - 4944275 p^{6} T^{19} + 849664 p^{7} T^{20} - 132180 p^{8} T^{21} + 18426 p^{9} T^{22} - 2230 p^{10} T^{23} + 229 p^{11} T^{24} - 18 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 - 3 T + 123 T^{2} - 466 T^{3} + 8032 T^{4} - 32730 T^{5} + 361778 T^{6} - 1461553 T^{7} + 12226232 T^{8} - 47067853 T^{9} + 322825250 T^{10} - 1154302277 T^{11} + 6808471240 T^{12} - 22113367388 T^{13} + 6808471240 p T^{14} - 1154302277 p^{2} T^{15} + 322825250 p^{3} T^{16} - 47067853 p^{4} T^{17} + 12226232 p^{5} T^{18} - 1461553 p^{6} T^{19} + 361778 p^{7} T^{20} - 32730 p^{8} T^{21} + 8032 p^{9} T^{22} - 466 p^{10} T^{23} + 123 p^{11} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 - 8 T + 150 T^{2} - 995 T^{3} + 11176 T^{4} - 2804 p T^{5} + 564008 T^{6} - 2919675 T^{7} + 21809092 T^{8} - 103215072 T^{9} + 688580607 T^{10} - 3011745618 T^{11} + 18401255144 T^{12} - 74730405848 T^{13} + 18401255144 p T^{14} - 3011745618 p^{2} T^{15} + 688580607 p^{3} T^{16} - 103215072 p^{4} T^{17} + 21809092 p^{5} T^{18} - 2919675 p^{6} T^{19} + 564008 p^{7} T^{20} - 2804 p^{9} T^{21} + 11176 p^{9} T^{22} - 995 p^{10} T^{23} + 150 p^{11} T^{24} - 8 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 - 11 T + 293 T^{2} - 2706 T^{3} + 40193 T^{4} - 321951 T^{5} + 120064 p T^{6} - 24653939 T^{7} + 7411844 p T^{8} - 1359010431 T^{9} + 10051369346 T^{10} - 56946727427 T^{11} + 367587386791 T^{12} - 1862125179534 T^{13} + 367587386791 p T^{14} - 56946727427 p^{2} T^{15} + 10051369346 p^{3} T^{16} - 1359010431 p^{4} T^{17} + 7411844 p^{6} T^{18} - 24653939 p^{6} T^{19} + 120064 p^{8} T^{20} - 321951 p^{8} T^{21} + 40193 p^{9} T^{22} - 2706 p^{10} T^{23} + 293 p^{11} T^{24} - 11 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 - 2 T + 199 T^{2} - 327 T^{3} + 17769 T^{4} - 23857 T^{5} + 969035 T^{6} - 1132915 T^{7} + 38927187 T^{8} - 47515578 T^{9} + 44236211 p T^{10} - 63631898 p T^{11} + 46011033826 T^{12} - 69587930342 T^{13} + 46011033826 p T^{14} - 63631898 p^{3} T^{15} + 44236211 p^{4} T^{16} - 47515578 p^{4} T^{17} + 38927187 p^{5} T^{18} - 1132915 p^{6} T^{19} + 969035 p^{7} T^{20} - 23857 p^{8} T^{21} + 17769 p^{9} T^{22} - 327 p^{10} T^{23} + 199 p^{11} T^{24} - 2 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 - 26 T + 529 T^{2} - 7904 T^{3} + 103262 T^{4} - 1169056 T^{5} + 12019094 T^{6} - 112358959 T^{7} + 973023469 T^{8} - 7821880362 T^{9} + 58927065033 T^{10} - 415931966177 T^{11} + 2767794504844 T^{12} - 17322216197496 T^{13} + 2767794504844 p T^{14} - 415931966177 p^{2} T^{15} + 58927065033 p^{3} T^{16} - 7821880362 p^{4} T^{17} + 973023469 p^{5} T^{18} - 112358959 p^{6} T^{19} + 12019094 p^{7} T^{20} - 1169056 p^{8} T^{21} + 103262 p^{9} T^{22} - 7904 p^{10} T^{23} + 529 p^{11} T^{24} - 26 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 3 T + 257 T^{2} - 1467 T^{3} + 34740 T^{4} - 253508 T^{5} + 3491069 T^{6} - 25669713 T^{7} + 278109082 T^{8} - 1883021097 T^{9} + 17430580176 T^{10} - 108869351172 T^{11} + 872814080755 T^{12} - 5013746853344 T^{13} + 872814080755 p T^{14} - 108869351172 p^{2} T^{15} + 17430580176 p^{3} T^{16} - 1883021097 p^{4} T^{17} + 278109082 p^{5} T^{18} - 25669713 p^{6} T^{19} + 3491069 p^{7} T^{20} - 253508 p^{8} T^{21} + 34740 p^{9} T^{22} - 1467 p^{10} T^{23} + 257 p^{11} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 - 24 T + 530 T^{2} - 7595 T^{3} + 104532 T^{4} - 1156702 T^{5} + 12558348 T^{6} - 117673007 T^{7} + 1088298600 T^{8} - 8950722176 T^{9} + 72757627443 T^{10} - 533590726990 T^{11} + 3871341804992 T^{12} - 25513634843988 T^{13} + 3871341804992 p T^{14} - 533590726990 p^{2} T^{15} + 72757627443 p^{3} T^{16} - 8950722176 p^{4} T^{17} + 1088298600 p^{5} T^{18} - 117673007 p^{6} T^{19} + 12558348 p^{7} T^{20} - 1156702 p^{8} T^{21} + 104532 p^{9} T^{22} - 7595 p^{10} T^{23} + 530 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 - 4 T + 268 T^{2} - 1033 T^{3} + 36836 T^{4} - 2600 p T^{5} + 3395347 T^{6} - 8423020 T^{7} + 234170305 T^{8} - 353238168 T^{9} + 13083450776 T^{10} - 8602214943 T^{11} + 645830400529 T^{12} - 191333851936 T^{13} + 645830400529 p T^{14} - 8602214943 p^{2} T^{15} + 13083450776 p^{3} T^{16} - 353238168 p^{4} T^{17} + 234170305 p^{5} T^{18} - 8423020 p^{6} T^{19} + 3395347 p^{7} T^{20} - 2600 p^{9} T^{21} + 36836 p^{9} T^{22} - 1033 p^{10} T^{23} + 268 p^{11} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 - 8 T + 488 T^{2} - 4056 T^{3} + 1978 p T^{4} - 955302 T^{5} + 18233248 T^{6} - 141093644 T^{7} + 2084160979 T^{8} - 14904118150 T^{9} + 185090156460 T^{10} - 1213436893580 T^{11} + 13261524523706 T^{12} - 79351524183736 T^{13} + 13261524523706 p T^{14} - 1213436893580 p^{2} T^{15} + 185090156460 p^{3} T^{16} - 14904118150 p^{4} T^{17} + 2084160979 p^{5} T^{18} - 141093644 p^{6} T^{19} + 18233248 p^{7} T^{20} - 955302 p^{8} T^{21} + 1978 p^{10} T^{22} - 4056 p^{10} T^{23} + 488 p^{11} T^{24} - 8 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 - 29 T + 719 T^{2} - 12106 T^{3} + 182257 T^{4} - 2246365 T^{5} + 25885890 T^{6} - 263725067 T^{7} + 2623379820 T^{8} - 24187988835 T^{9} + 222738683596 T^{10} - 1921212995771 T^{11} + 16380793423473 T^{12} - 129356675228598 T^{13} + 16380793423473 p T^{14} - 1921212995771 p^{2} T^{15} + 222738683596 p^{3} T^{16} - 24187988835 p^{4} T^{17} + 2623379820 p^{5} T^{18} - 263725067 p^{6} T^{19} + 25885890 p^{7} T^{20} - 2246365 p^{8} T^{21} + 182257 p^{9} T^{22} - 12106 p^{10} T^{23} + 719 p^{11} T^{24} - 29 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 + 5 T + 479 T^{2} + 2461 T^{3} + 113540 T^{4} + 565244 T^{5} + 17806940 T^{6} + 82160097 T^{7} + 2083232523 T^{8} + 8672645107 T^{9} + 194767833389 T^{10} + 729094736874 T^{11} + 15225633883928 T^{12} + 52217255381240 T^{13} + 15225633883928 p T^{14} + 729094736874 p^{2} T^{15} + 194767833389 p^{3} T^{16} + 8672645107 p^{4} T^{17} + 2083232523 p^{5} T^{18} + 82160097 p^{6} T^{19} + 17806940 p^{7} T^{20} + 565244 p^{8} T^{21} + 113540 p^{9} T^{22} + 2461 p^{10} T^{23} + 479 p^{11} T^{24} + 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 - 29 T + 720 T^{2} - 12572 T^{3} + 193566 T^{4} - 2524156 T^{5} + 30049566 T^{6} - 321700200 T^{7} + 3227480510 T^{8} - 30090521795 T^{9} + 270116549823 T^{10} - 2324136264700 T^{11} + 19845964258776 T^{12} - 166590249768584 T^{13} + 19845964258776 p T^{14} - 2324136264700 p^{2} T^{15} + 270116549823 p^{3} T^{16} - 30090521795 p^{4} T^{17} + 3227480510 p^{5} T^{18} - 321700200 p^{6} T^{19} + 30049566 p^{7} T^{20} - 2524156 p^{8} T^{21} + 193566 p^{9} T^{22} - 12572 p^{10} T^{23} + 720 p^{11} T^{24} - 29 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 15 T + 557 T^{2} - 8027 T^{3} + 167520 T^{4} - 2162404 T^{5} + 33852096 T^{6} - 389448255 T^{7} + 5033721195 T^{8} - 51826456265 T^{9} + 578222783383 T^{10} - 5345218355286 T^{11} + 52641732870928 T^{12} - 436897445699016 T^{13} + 52641732870928 p T^{14} - 5345218355286 p^{2} T^{15} + 578222783383 p^{3} T^{16} - 51826456265 p^{4} T^{17} + 5033721195 p^{5} T^{18} - 389448255 p^{6} T^{19} + 33852096 p^{7} T^{20} - 2162404 p^{8} T^{21} + 167520 p^{9} T^{22} - 8027 p^{10} T^{23} + 557 p^{11} T^{24} - 15 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 - 15 T + 655 T^{2} - 7729 T^{3} + 200993 T^{4} - 1936471 T^{5} + 38812327 T^{6} - 310349774 T^{7} + 5363658003 T^{8} - 36115904519 T^{9} + 577889673817 T^{10} - 3372883398881 T^{11} + 52084492714874 T^{12} - 277652174402022 T^{13} + 52084492714874 p T^{14} - 3372883398881 p^{2} T^{15} + 577889673817 p^{3} T^{16} - 36115904519 p^{4} T^{17} + 5363658003 p^{5} T^{18} - 310349774 p^{6} T^{19} + 38812327 p^{7} T^{20} - 1936471 p^{8} T^{21} + 200993 p^{9} T^{22} - 7729 p^{10} T^{23} + 655 p^{11} T^{24} - 15 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 - 4 T + 473 T^{2} - 1122 T^{3} + 107130 T^{4} - 71316 T^{5} + 15994894 T^{6} + 17009403 T^{7} + 1826527295 T^{8} + 4796438420 T^{9} + 176988246491 T^{10} + 654794730599 T^{11} + 15654692771376 T^{12} + 62609920158744 T^{13} + 15654692771376 p T^{14} + 654794730599 p^{2} T^{15} + 176988246491 p^{3} T^{16} + 4796438420 p^{4} T^{17} + 1826527295 p^{5} T^{18} + 17009403 p^{6} T^{19} + 15994894 p^{7} T^{20} - 71316 p^{8} T^{21} + 107130 p^{9} T^{22} - 1122 p^{10} T^{23} + 473 p^{11} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 3 T + 446 T^{2} - 1828 T^{3} + 98490 T^{4} - 515170 T^{5} + 14334284 T^{6} - 99999528 T^{7} + 1555706438 T^{8} - 15158087141 T^{9} + 138708660035 T^{10} - 1848037464820 T^{11} + 11570893576074 T^{12} - 182365098822956 T^{13} + 11570893576074 p T^{14} - 1848037464820 p^{2} T^{15} + 138708660035 p^{3} T^{16} - 15158087141 p^{4} T^{17} + 1555706438 p^{5} T^{18} - 99999528 p^{6} T^{19} + 14334284 p^{7} T^{20} - 515170 p^{8} T^{21} + 98490 p^{9} T^{22} - 1828 p^{10} T^{23} + 446 p^{11} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 - 50 T + 1882 T^{2} - 50535 T^{3} + 1142892 T^{4} - 21649886 T^{5} + 362096140 T^{6} - 5357588155 T^{7} + 72052590140 T^{8} - 885465616422 T^{9} + 10155880272401 T^{10} - 109630009836830 T^{11} + 11704168726528 p T^{12} - 11336309613424532 T^{13} + 11704168726528 p^{2} T^{14} - 109630009836830 p^{2} T^{15} + 10155880272401 p^{3} T^{16} - 885465616422 p^{4} T^{17} + 72052590140 p^{5} T^{18} - 5357588155 p^{6} T^{19} + 362096140 p^{7} T^{20} - 21649886 p^{8} T^{21} + 1142892 p^{9} T^{22} - 50535 p^{10} T^{23} + 1882 p^{11} T^{24} - 50 p^{12} T^{25} + p^{13} T^{26} \) | |
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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
−2.28153282007440726247967108526, −2.20336744616815555805788189151, −2.19388855197284914324859507633, −2.11679605850612478447534576307, −2.09539654815201861366200395790, −2.06439459921866912706553756385, −1.98315443212970056184690508635, −1.98230486574382847105663398831, −1.90057387207596089024532403107, −1.86086969718933768907321439587, −1.81966775083111908662447333793, −1.77401049007537515370807728196, −1.37409726128445129593198700170, −1.35364538412557692515924861690, −1.29363436475625001497779638042, −1.26980586800364043571209726999, −1.22294653884838350817029885539, −1.16178347118396546788925657239, −1.13627290177184954226783329147, −1.02692830127875923300312132109, −0.953089253865494263116087313569, −0.861597328478426437777696960375, −0.831927019364024019080600676987, −0.827405543399686897923846245869, −0.62924052880461913865842469260, 0.62924052880461913865842469260, 0.827405543399686897923846245869, 0.831927019364024019080600676987, 0.861597328478426437777696960375, 0.953089253865494263116087313569, 1.02692830127875923300312132109, 1.13627290177184954226783329147, 1.16178347118396546788925657239, 1.22294653884838350817029885539, 1.26980586800364043571209726999, 1.29363436475625001497779638042, 1.35364538412557692515924861690, 1.37409726128445129593198700170, 1.77401049007537515370807728196, 1.81966775083111908662447333793, 1.86086969718933768907321439587, 1.90057387207596089024532403107, 1.98230486574382847105663398831, 1.98315443212970056184690508635, 2.06439459921866912706553756385, 2.09539654815201861366200395790, 2.11679605850612478447534576307, 2.19388855197284914324859507633, 2.20336744616815555805788189151, 2.28153282007440726247967108526
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.