Dirichlet series
| L(s) = 1 | + 13·5-s − 13·7-s − 12·9-s + 11·13-s + 16·17-s + 3·23-s + 91·25-s + 2·27-s + 10·29-s − 31-s − 169·35-s + 16·37-s + 23·41-s + 13·43-s − 156·45-s + 2·47-s + 91·49-s + 20·53-s + 2·59-s + 5·61-s + 156·63-s + 143·65-s + 19·67-s + 4·71-s + 34·73-s − 15·79-s + 65·81-s + ⋯ |
| L(s) = 1 | + 5.81·5-s − 4.91·7-s − 4·9-s + 3.05·13-s + 3.88·17-s + 0.625·23-s + 91/5·25-s + 0.384·27-s + 1.85·29-s − 0.179·31-s − 28.5·35-s + 2.63·37-s + 3.59·41-s + 1.98·43-s − 23.2·45-s + 0.291·47-s + 13·49-s + 2.74·53-s + 0.260·59-s + 0.640·61-s + 19.6·63-s + 17.7·65-s + 2.32·67-s + 0.474·71-s + 3.97·73-s − 1.68·79-s + 65/9·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{26} \cdot 5^{13} \cdot 7^{13} \cdot 43^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{26} \cdot 5^{13} \cdot 7^{13} \cdot 43^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(2^{26} \cdot 5^{13} \cdot 7^{13} \cdot 43^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.31739\times 10^{21}\) |
| Root analytic conductor: | \(6.93324\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 2^{26} \cdot 5^{13} \cdot 7^{13} \cdot 43^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(909.0184776\) |
| \(L(\frac12)\) | \(\approx\) | \(909.0184776\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T )^{13} \) | |
| 7 | \( ( 1 + T )^{13} \) | |
| 43 | \( ( 1 - T )^{13} \) | |
| good | 3 | \( 1 + 4 p T^{2} - 2 T^{3} + 79 T^{4} - 23 T^{5} + 392 T^{6} - 37 p T^{7} + 1625 T^{8} - 109 p T^{9} + 5897 T^{10} - 799 T^{11} + 6383 p T^{12} - 2152 T^{13} + 6383 p^{2} T^{14} - 799 p^{2} T^{15} + 5897 p^{3} T^{16} - 109 p^{5} T^{17} + 1625 p^{5} T^{18} - 37 p^{7} T^{19} + 392 p^{7} T^{20} - 23 p^{8} T^{21} + 79 p^{9} T^{22} - 2 p^{10} T^{23} + 4 p^{12} T^{24} + p^{13} T^{26} \) |
| 11 | \( 1 + 53 T^{2} + 50 T^{3} + 1501 T^{4} + 2028 T^{5} + 30139 T^{6} + 49789 T^{7} + 460566 T^{8} + 887374 T^{9} + 5984420 T^{10} + 12445289 T^{11} + 69492496 T^{12} + 149524172 T^{13} + 69492496 p T^{14} + 12445289 p^{2} T^{15} + 5984420 p^{3} T^{16} + 887374 p^{4} T^{17} + 460566 p^{5} T^{18} + 49789 p^{6} T^{19} + 30139 p^{7} T^{20} + 2028 p^{8} T^{21} + 1501 p^{9} T^{22} + 50 p^{10} T^{23} + 53 p^{11} T^{24} + p^{13} T^{26} \) | |
| 13 | \( 1 - 11 T + 124 T^{2} - 925 T^{3} + 489 p T^{4} - 36721 T^{5} + 195108 T^{6} - 942729 T^{7} + 4270443 T^{8} - 18191133 T^{9} + 74302953 T^{10} - 289333586 T^{11} + 1099364059 T^{12} - 3999583790 T^{13} + 1099364059 p T^{14} - 289333586 p^{2} T^{15} + 74302953 p^{3} T^{16} - 18191133 p^{4} T^{17} + 4270443 p^{5} T^{18} - 942729 p^{6} T^{19} + 195108 p^{7} T^{20} - 36721 p^{8} T^{21} + 489 p^{10} T^{22} - 925 p^{10} T^{23} + 124 p^{11} T^{24} - 11 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 - 16 T + 232 T^{2} - 2164 T^{3} + 18346 T^{4} - 123150 T^{5} + 772393 T^{6} - 4181271 T^{7} + 21988401 T^{8} - 105957487 T^{9} + 510803344 T^{10} - 2309289270 T^{11} + 10360350472 T^{12} - 43159109488 T^{13} + 10360350472 p T^{14} - 2309289270 p^{2} T^{15} + 510803344 p^{3} T^{16} - 105957487 p^{4} T^{17} + 21988401 p^{5} T^{18} - 4181271 p^{6} T^{19} + 772393 p^{7} T^{20} - 123150 p^{8} T^{21} + 18346 p^{9} T^{22} - 2164 p^{10} T^{23} + 232 p^{11} T^{24} - 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 + 117 T^{2} + 5 T^{3} + 6835 T^{4} - 997 T^{5} + 266583 T^{6} - 134458 T^{7} + 7847403 T^{8} - 7348506 T^{9} + 188460095 T^{10} - 243708059 T^{11} + 3937295160 T^{12} - 5517755554 T^{13} + 3937295160 p T^{14} - 243708059 p^{2} T^{15} + 188460095 p^{3} T^{16} - 7348506 p^{4} T^{17} + 7847403 p^{5} T^{18} - 134458 p^{6} T^{19} + 266583 p^{7} T^{20} - 997 p^{8} T^{21} + 6835 p^{9} T^{22} + 5 p^{10} T^{23} + 117 p^{11} T^{24} + p^{13} T^{26} \) | |
| 23 | \( 1 - 3 T + 151 T^{2} - 443 T^{3} + 11161 T^{4} - 32060 T^{5} + 548870 T^{6} - 1558229 T^{7} + 20629834 T^{8} - 58509997 T^{9} + 639250817 T^{10} - 1798029221 T^{11} + 16949966784 T^{12} - 45597290078 T^{13} + 16949966784 p T^{14} - 1798029221 p^{2} T^{15} + 639250817 p^{3} T^{16} - 58509997 p^{4} T^{17} + 20629834 p^{5} T^{18} - 1558229 p^{6} T^{19} + 548870 p^{7} T^{20} - 32060 p^{8} T^{21} + 11161 p^{9} T^{22} - 443 p^{10} T^{23} + 151 p^{11} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 - 10 T + 238 T^{2} - 2079 T^{3} + 28142 T^{4} - 215709 T^{5} + 2184543 T^{6} - 14852326 T^{7} + 124454688 T^{8} - 758718278 T^{9} + 5514778387 T^{10} - 30368027851 T^{11} + 196241084625 T^{12} - 977916685190 T^{13} + 196241084625 p T^{14} - 30368027851 p^{2} T^{15} + 5514778387 p^{3} T^{16} - 758718278 p^{4} T^{17} + 124454688 p^{5} T^{18} - 14852326 p^{6} T^{19} + 2184543 p^{7} T^{20} - 215709 p^{8} T^{21} + 28142 p^{9} T^{22} - 2079 p^{10} T^{23} + 238 p^{11} T^{24} - 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 + T + 88 T^{2} + 108 T^{3} + 5936 T^{4} + 5938 T^{5} + 335461 T^{6} + 258870 T^{7} + 14930920 T^{8} + 9479568 T^{9} + 590741078 T^{10} + 294357379 T^{11} + 20604001480 T^{12} + 8929202624 T^{13} + 20604001480 p T^{14} + 294357379 p^{2} T^{15} + 590741078 p^{3} T^{16} + 9479568 p^{4} T^{17} + 14930920 p^{5} T^{18} + 258870 p^{6} T^{19} + 335461 p^{7} T^{20} + 5938 p^{8} T^{21} + 5936 p^{9} T^{22} + 108 p^{10} T^{23} + 88 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 - 16 T + 335 T^{2} - 3583 T^{3} + 47783 T^{4} - 421511 T^{5} + 4504539 T^{6} - 34659239 T^{7} + 316406640 T^{8} - 2174885757 T^{9} + 17559909034 T^{10} - 109093513955 T^{11} + 790552591108 T^{12} - 4453783675410 T^{13} + 790552591108 p T^{14} - 109093513955 p^{2} T^{15} + 17559909034 p^{3} T^{16} - 2174885757 p^{4} T^{17} + 316406640 p^{5} T^{18} - 34659239 p^{6} T^{19} + 4504539 p^{7} T^{20} - 421511 p^{8} T^{21} + 47783 p^{9} T^{22} - 3583 p^{10} T^{23} + 335 p^{11} T^{24} - 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 23 T + 576 T^{2} - 9070 T^{3} + 139315 T^{4} - 1704579 T^{5} + 19979305 T^{6} - 201725431 T^{7} + 47492324 p T^{8} - 16761244471 T^{9} + 138092320179 T^{10} - 1031759307419 T^{11} + 7387434553068 T^{12} - 48308950864830 T^{13} + 7387434553068 p T^{14} - 1031759307419 p^{2} T^{15} + 138092320179 p^{3} T^{16} - 16761244471 p^{4} T^{17} + 47492324 p^{6} T^{18} - 201725431 p^{6} T^{19} + 19979305 p^{7} T^{20} - 1704579 p^{8} T^{21} + 139315 p^{9} T^{22} - 9070 p^{10} T^{23} + 576 p^{11} T^{24} - 23 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 - 2 T + 284 T^{2} - 299 T^{3} + 41950 T^{4} - 5019 T^{5} + 4220701 T^{6} + 3181572 T^{7} + 326022532 T^{8} + 474687616 T^{9} + 20670190075 T^{10} + 38654405367 T^{11} + 23796987455 p T^{12} + 2149968933466 T^{13} + 23796987455 p^{2} T^{14} + 38654405367 p^{2} T^{15} + 20670190075 p^{3} T^{16} + 474687616 p^{4} T^{17} + 326022532 p^{5} T^{18} + 3181572 p^{6} T^{19} + 4220701 p^{7} T^{20} - 5019 p^{8} T^{21} + 41950 p^{9} T^{22} - 299 p^{10} T^{23} + 284 p^{11} T^{24} - 2 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 - 20 T + 570 T^{2} - 8239 T^{3} + 140200 T^{4} - 1616147 T^{5} + 20795049 T^{6} - 201364088 T^{7} + 2145810135 T^{8} - 18082107742 T^{9} + 168072325522 T^{10} - 1267913455673 T^{11} + 10633847752991 T^{12} - 73271137966998 T^{13} + 10633847752991 p T^{14} - 1267913455673 p^{2} T^{15} + 168072325522 p^{3} T^{16} - 18082107742 p^{4} T^{17} + 2145810135 p^{5} T^{18} - 201364088 p^{6} T^{19} + 20795049 p^{7} T^{20} - 1616147 p^{8} T^{21} + 140200 p^{9} T^{22} - 8239 p^{10} T^{23} + 570 p^{11} T^{24} - 20 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 - 2 T + 336 T^{2} + 278 T^{3} + 57047 T^{4} + 172755 T^{5} + 6988059 T^{6} + 30304986 T^{7} + 699474600 T^{8} + 3308404742 T^{9} + 58857012758 T^{10} + 270061385137 T^{11} + 4148204927199 T^{12} + 17669714125088 T^{13} + 4148204927199 p T^{14} + 270061385137 p^{2} T^{15} + 58857012758 p^{3} T^{16} + 3308404742 p^{4} T^{17} + 699474600 p^{5} T^{18} + 30304986 p^{6} T^{19} + 6988059 p^{7} T^{20} + 172755 p^{8} T^{21} + 57047 p^{9} T^{22} + 278 p^{10} T^{23} + 336 p^{11} T^{24} - 2 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 - 5 T + 385 T^{2} - 1392 T^{3} + 78249 T^{4} - 228619 T^{5} + 11220912 T^{6} - 27733767 T^{7} + 1238659618 T^{8} - 2654743135 T^{9} + 110130687753 T^{10} - 209812101594 T^{11} + 8071813843508 T^{12} - 13922354346548 T^{13} + 8071813843508 p T^{14} - 209812101594 p^{2} T^{15} + 110130687753 p^{3} T^{16} - 2654743135 p^{4} T^{17} + 1238659618 p^{5} T^{18} - 27733767 p^{6} T^{19} + 11220912 p^{7} T^{20} - 228619 p^{8} T^{21} + 78249 p^{9} T^{22} - 1392 p^{10} T^{23} + 385 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 - 19 T + 665 T^{2} - 10361 T^{3} + 214846 T^{4} - 2819094 T^{5} + 44046119 T^{6} - 499516783 T^{7} + 6414440638 T^{8} - 63876028473 T^{9} + 702928421016 T^{10} - 6203811416389 T^{11} + 59811288016869 T^{12} - 469196028137730 T^{13} + 59811288016869 p T^{14} - 6203811416389 p^{2} T^{15} + 702928421016 p^{3} T^{16} - 63876028473 p^{4} T^{17} + 6414440638 p^{5} T^{18} - 499516783 p^{6} T^{19} + 44046119 p^{7} T^{20} - 2819094 p^{8} T^{21} + 214846 p^{9} T^{22} - 10361 p^{10} T^{23} + 665 p^{11} T^{24} - 19 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 - 4 T + 548 T^{2} - 2789 T^{3} + 147676 T^{4} - 863499 T^{5} + 26477214 T^{6} - 164426873 T^{7} + 3562074814 T^{8} - 22118088775 T^{9} + 379875465159 T^{10} - 2254879842485 T^{11} + 32902726454248 T^{12} - 180052027863694 T^{13} + 32902726454248 p T^{14} - 2254879842485 p^{2} T^{15} + 379875465159 p^{3} T^{16} - 22118088775 p^{4} T^{17} + 3562074814 p^{5} T^{18} - 164426873 p^{6} T^{19} + 26477214 p^{7} T^{20} - 863499 p^{8} T^{21} + 147676 p^{9} T^{22} - 2789 p^{10} T^{23} + 548 p^{11} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 34 T + 1121 T^{2} - 25315 T^{3} + 516240 T^{4} - 8885498 T^{5} + 138834580 T^{6} - 1945250236 T^{7} + 25048831315 T^{8} - 296064712316 T^{9} + 3247541880683 T^{10} - 33025369477201 T^{11} + 313501550116924 T^{12} - 2769720931551088 T^{13} + 313501550116924 p T^{14} - 33025369477201 p^{2} T^{15} + 3247541880683 p^{3} T^{16} - 296064712316 p^{4} T^{17} + 25048831315 p^{5} T^{18} - 1945250236 p^{6} T^{19} + 138834580 p^{7} T^{20} - 8885498 p^{8} T^{21} + 516240 p^{9} T^{22} - 25315 p^{10} T^{23} + 1121 p^{11} T^{24} - 34 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 + 15 T + 443 T^{2} + 5563 T^{3} + 99432 T^{4} + 962780 T^{5} + 12890577 T^{6} + 90762914 T^{7} + 956871031 T^{8} + 2892970095 T^{9} + 23658872691 T^{10} - 363925121613 T^{11} - 2458340352577 T^{12} - 52611160040276 T^{13} - 2458340352577 p T^{14} - 363925121613 p^{2} T^{15} + 23658872691 p^{3} T^{16} + 2892970095 p^{4} T^{17} + 956871031 p^{5} T^{18} + 90762914 p^{6} T^{19} + 12890577 p^{7} T^{20} + 962780 p^{8} T^{21} + 99432 p^{9} T^{22} + 5563 p^{10} T^{23} + 443 p^{11} T^{24} + 15 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 - 27 T + 934 T^{2} - 17473 T^{3} + 368115 T^{4} - 5500034 T^{5} + 89024957 T^{6} - 1128150732 T^{7} + 15245376228 T^{8} - 169102543191 T^{9} + 1986785395163 T^{10} - 19626628915027 T^{11} + 204813701011982 T^{12} - 1816049205194936 T^{13} + 204813701011982 p T^{14} - 19626628915027 p^{2} T^{15} + 1986785395163 p^{3} T^{16} - 169102543191 p^{4} T^{17} + 15245376228 p^{5} T^{18} - 1128150732 p^{6} T^{19} + 89024957 p^{7} T^{20} - 5500034 p^{8} T^{21} + 368115 p^{9} T^{22} - 17473 p^{10} T^{23} + 934 p^{11} T^{24} - 27 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 3 T + 525 T^{2} - 2921 T^{3} + 144772 T^{4} - 1059644 T^{5} + 28951997 T^{6} - 228043921 T^{7} + 4615052546 T^{8} - 35443806123 T^{9} + 597964680032 T^{10} - 4340614081579 T^{11} + 63785482146699 T^{12} - 429326946412826 T^{13} + 63785482146699 p T^{14} - 4340614081579 p^{2} T^{15} + 597964680032 p^{3} T^{16} - 35443806123 p^{4} T^{17} + 4615052546 p^{5} T^{18} - 228043921 p^{6} T^{19} + 28951997 p^{7} T^{20} - 1059644 p^{8} T^{21} + 144772 p^{9} T^{22} - 2921 p^{10} T^{23} + 525 p^{11} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 - 45 T + 1601 T^{2} - 38478 T^{3} + 793352 T^{4} - 13225379 T^{5} + 198615715 T^{6} - 2604983062 T^{7} + 32405292753 T^{8} - 374378589486 T^{9} + 4279452478888 T^{10} - 46557894756050 T^{11} + 497998439882051 T^{12} - 4986376893479044 T^{13} + 497998439882051 p T^{14} - 46557894756050 p^{2} T^{15} + 4279452478888 p^{3} T^{16} - 374378589486 p^{4} T^{17} + 32405292753 p^{5} T^{18} - 2604983062 p^{6} T^{19} + 198615715 p^{7} T^{20} - 13225379 p^{8} T^{21} + 793352 p^{9} T^{22} - 38478 p^{10} T^{23} + 1601 p^{11} T^{24} - 45 p^{12} T^{25} + p^{13} T^{26} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.11572377364313355960879342775, −2.08029422991873505290551525890, −2.07849221787683279062059585039, −2.07310859289867794931611583160, −1.98731837502798610611836037477, −1.98000923523882487811281534133, −1.86055515932839958740818456501, −1.83519707162927328235493615457, −1.77249456754199532216471939346, −1.61538711820888771668159744868, −1.52153332790694336839578843989, −1.28362872993812253685270950275, −1.09545612379415446167895979903, −1.08401558857842659286421045034, −1.00769135798659235571333932233, −0.919731065444282403672209422878, −0.911667581509630873612694761413, −0.850781597888523430588944917524, −0.75360729375673409977095007292, −0.69303712453876610535281809138, −0.60858313079387383690419762832, −0.56894505114254834563048223934, −0.56009042570253486439687293466, −0.48819583691021731855957939764, −0.23191403349461611010384018304, 0.23191403349461611010384018304, 0.48819583691021731855957939764, 0.56009042570253486439687293466, 0.56894505114254834563048223934, 0.60858313079387383690419762832, 0.69303712453876610535281809138, 0.75360729375673409977095007292, 0.850781597888523430588944917524, 0.911667581509630873612694761413, 0.919731065444282403672209422878, 1.00769135798659235571333932233, 1.08401558857842659286421045034, 1.09545612379415446167895979903, 1.28362872993812253685270950275, 1.52153332790694336839578843989, 1.61538711820888771668159744868, 1.77249456754199532216471939346, 1.83519707162927328235493615457, 1.86055515932839958740818456501, 1.98000923523882487811281534133, 1.98731837502798610611836037477, 2.07310859289867794931611583160, 2.07849221787683279062059585039, 2.08029422991873505290551525890, 2.11572377364313355960879342775
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.