Dirichlet series
| L(s) = 1 | + 4·5-s + 4·11-s − 2·13-s + 2·17-s − 7·19-s + 22·23-s − 18·25-s − 29-s + 31-s + 10·37-s − 33·41-s + 7·43-s − 3·47-s + 15·53-s + 16·55-s − 14·59-s + 10·61-s − 8·65-s + 6·67-s − 2·71-s − 21·73-s − 10·79-s − 25·83-s + 8·85-s − 6·89-s − 28·95-s + 18·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 1.60·19-s + 4.58·23-s − 3.59·25-s − 0.185·29-s + 0.179·31-s + 1.64·37-s − 5.15·41-s + 1.06·43-s − 0.437·47-s + 2.06·53-s + 2.15·55-s − 1.82·59-s + 1.28·61-s − 0.992·65-s + 0.733·67-s − 0.237·71-s − 2.45·73-s − 1.12·79-s − 2.74·83-s + 0.867·85-s − 0.635·89-s − 2.87·95-s + 1.82·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{42} \cdot 7^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{42} \cdot 7^{28}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{28} \cdot 3^{42} \cdot 7^{28}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.78838\times 10^{22}\) |
| Root analytic conductor: | \(6.50052\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{28} \cdot 3^{42} \cdot 7^{28} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.171858201\) |
| \(L(\frac12)\) | \(\approx\) | \(4.171858201\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( ( 1 - 2 T + 3 p T^{2} - 48 T^{3} + 154 T^{4} - 429 T^{5} + 1202 T^{6} - 2471 T^{7} + 1202 p T^{8} - 429 p^{2} T^{9} + 154 p^{3} T^{10} - 48 p^{4} T^{11} + 3 p^{6} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) |
| 11 | \( ( 1 - 2 T + 36 T^{2} - 57 T^{3} + 460 T^{4} - 402 T^{5} + 2501 T^{6} - 455 T^{7} + 2501 p T^{8} - 402 p^{2} T^{9} + 460 p^{3} T^{10} - 57 p^{4} T^{11} + 36 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 13 | \( 1 + 2 T - 25 T^{2} + 20 T^{3} + 172 T^{4} - 1281 T^{5} + 1882 T^{6} + 1142 T^{7} - 27931 T^{8} + 309997 T^{9} + 9092 T^{10} - 1859069 T^{11} + 348475 p T^{12} - 5185603 T^{13} - 73974588 T^{14} - 5185603 p T^{15} + 348475 p^{3} T^{16} - 1859069 p^{3} T^{17} + 9092 p^{4} T^{18} + 309997 p^{5} T^{19} - 27931 p^{6} T^{20} + 1142 p^{7} T^{21} + 1882 p^{8} T^{22} - 1281 p^{9} T^{23} + 172 p^{10} T^{24} + 20 p^{11} T^{25} - 25 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 - 2 T - 65 T^{2} + 210 T^{3} + 2087 T^{4} - 9143 T^{5} - 37340 T^{6} + 240381 T^{7} + 293834 T^{8} - 4176065 T^{9} + 3382763 T^{10} + 47662233 T^{11} - 157347285 T^{12} - 273515658 T^{13} + 3170983122 T^{14} - 273515658 p T^{15} - 157347285 p^{2} T^{16} + 47662233 p^{3} T^{17} + 3382763 p^{4} T^{18} - 4176065 p^{5} T^{19} + 293834 p^{6} T^{20} + 240381 p^{7} T^{21} - 37340 p^{8} T^{22} - 9143 p^{9} T^{23} + 2087 p^{10} T^{24} + 210 p^{11} T^{25} - 65 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 19 | \( 1 + 7 T - 54 T^{2} - 381 T^{3} + 1875 T^{4} + 9873 T^{5} - 65652 T^{6} - 221430 T^{7} + 1870425 T^{8} + 4319703 T^{9} - 46476858 T^{10} - 61637031 T^{11} + 1073881146 T^{12} + 457871775 T^{13} - 21789737442 T^{14} + 457871775 p T^{15} + 1073881146 p^{2} T^{16} - 61637031 p^{3} T^{17} - 46476858 p^{4} T^{18} + 4319703 p^{5} T^{19} + 1870425 p^{6} T^{20} - 221430 p^{7} T^{21} - 65652 p^{8} T^{22} + 9873 p^{9} T^{23} + 1875 p^{10} T^{24} - 381 p^{11} T^{25} - 54 p^{12} T^{26} + 7 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( ( 1 - 11 T + 129 T^{2} - 876 T^{3} + 6604 T^{4} - 35691 T^{5} + 217568 T^{6} - 992939 T^{7} + 217568 p T^{8} - 35691 p^{2} T^{9} + 6604 p^{3} T^{10} - 876 p^{4} T^{11} + 129 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 29 | \( 1 + T - 89 T^{2} - 606 T^{3} + 3413 T^{4} + 45595 T^{5} + 49603 T^{6} - 1643802 T^{7} - 7893892 T^{8} + 19552444 T^{9} + 271585946 T^{10} + 420585531 T^{11} - 3441402105 T^{12} - 10956432363 T^{13} + 10513309734 T^{14} - 10956432363 p T^{15} - 3441402105 p^{2} T^{16} + 420585531 p^{3} T^{17} + 271585946 p^{4} T^{18} + 19552444 p^{5} T^{19} - 7893892 p^{6} T^{20} - 1643802 p^{7} T^{21} + 49603 p^{8} T^{22} + 45595 p^{9} T^{23} + 3413 p^{10} T^{24} - 606 p^{11} T^{25} - 89 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \) | |
| 31 | \( 1 - T - 85 T^{2} + 302 T^{3} + 2323 T^{4} - 16596 T^{5} - 8720 T^{6} + 121502 T^{7} + 336059 T^{8} + 5031700 T^{9} - 52734649 T^{10} + 249456712 T^{11} - 356372186 T^{12} - 8588183707 T^{13} + 82956488229 T^{14} - 8588183707 p T^{15} - 356372186 p^{2} T^{16} + 249456712 p^{3} T^{17} - 52734649 p^{4} T^{18} + 5031700 p^{5} T^{19} + 336059 p^{6} T^{20} + 121502 p^{7} T^{21} - 8720 p^{8} T^{22} - 16596 p^{9} T^{23} + 2323 p^{10} T^{24} + 302 p^{11} T^{25} - 85 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \) | |
| 37 | \( 1 - 10 T - 84 T^{2} + 1296 T^{3} + 1134 T^{4} - 66630 T^{5} + 77382 T^{6} + 1851174 T^{7} - 1251993 T^{8} - 21837810 T^{9} - 268575252 T^{10} - 774884982 T^{11} + 27666538119 T^{12} + 26159905206 T^{13} - 1381876923558 T^{14} + 26159905206 p T^{15} + 27666538119 p^{2} T^{16} - 774884982 p^{3} T^{17} - 268575252 p^{4} T^{18} - 21837810 p^{5} T^{19} - 1251993 p^{6} T^{20} + 1851174 p^{7} T^{21} + 77382 p^{8} T^{22} - 66630 p^{9} T^{23} + 1134 p^{10} T^{24} + 1296 p^{11} T^{25} - 84 p^{12} T^{26} - 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 41 | \( 1 + 33 T + 463 T^{2} + 3882 T^{3} + 26359 T^{4} + 177381 T^{5} + 987377 T^{6} + 3338436 T^{7} - 1489480 T^{8} - 146370792 T^{9} - 1532374696 T^{10} - 10553354379 T^{11} - 66691501599 T^{12} - 460808734581 T^{13} - 3110830401306 T^{14} - 460808734581 p T^{15} - 66691501599 p^{2} T^{16} - 10553354379 p^{3} T^{17} - 1532374696 p^{4} T^{18} - 146370792 p^{5} T^{19} - 1489480 p^{6} T^{20} + 3338436 p^{7} T^{21} + 987377 p^{8} T^{22} + 177381 p^{9} T^{23} + 26359 p^{10} T^{24} + 3882 p^{11} T^{25} + 463 p^{12} T^{26} + 33 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( 1 - 7 T - 222 T^{2} + 1221 T^{3} + 30003 T^{4} - 121065 T^{5} - 2964828 T^{6} + 8400318 T^{7} + 232132089 T^{8} - 430114695 T^{9} - 14993117802 T^{10} + 15328887375 T^{11} + 817956723570 T^{12} - 6029661837 p T^{13} - 884111517510 p T^{14} - 6029661837 p^{2} T^{15} + 817956723570 p^{2} T^{16} + 15328887375 p^{3} T^{17} - 14993117802 p^{4} T^{18} - 430114695 p^{5} T^{19} + 232132089 p^{6} T^{20} + 8400318 p^{7} T^{21} - 2964828 p^{8} T^{22} - 121065 p^{9} T^{23} + 30003 p^{10} T^{24} + 1221 p^{11} T^{25} - 222 p^{12} T^{26} - 7 p^{13} T^{27} + p^{14} T^{28} \) | |
| 47 | \( 1 + 3 T - 215 T^{2} - 750 T^{3} + 23197 T^{4} + 82998 T^{5} - 1793695 T^{6} - 5388753 T^{7} + 119401781 T^{8} + 227010657 T^{9} - 7448336422 T^{10} - 6436178352 T^{11} + 428559183834 T^{12} + 95001497937 T^{13} - 21681767924409 T^{14} + 95001497937 p T^{15} + 428559183834 p^{2} T^{16} - 6436178352 p^{3} T^{17} - 7448336422 p^{4} T^{18} + 227010657 p^{5} T^{19} + 119401781 p^{6} T^{20} - 5388753 p^{7} T^{21} - 1793695 p^{8} T^{22} + 82998 p^{9} T^{23} + 23197 p^{10} T^{24} - 750 p^{11} T^{25} - 215 p^{12} T^{26} + 3 p^{13} T^{27} + p^{14} T^{28} \) | |
| 53 | \( 1 - 15 T - 44 T^{2} + 1563 T^{3} + 1621 T^{4} - 132831 T^{5} + 280796 T^{6} + 6875916 T^{7} - 32544895 T^{8} - 334617081 T^{9} + 3332374934 T^{10} + 9404405181 T^{11} - 219667861866 T^{12} - 268474558113 T^{13} + 14089335458730 T^{14} - 268474558113 p T^{15} - 219667861866 p^{2} T^{16} + 9404405181 p^{3} T^{17} + 3332374934 p^{4} T^{18} - 334617081 p^{5} T^{19} - 32544895 p^{6} T^{20} + 6875916 p^{7} T^{21} + 280796 p^{8} T^{22} - 132831 p^{9} T^{23} + 1621 p^{10} T^{24} + 1563 p^{11} T^{25} - 44 p^{12} T^{26} - 15 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 + 14 T - 41 T^{2} - 1932 T^{3} - 9268 T^{4} + 67958 T^{5} + 1000462 T^{6} + 4853283 T^{7} - 9914932 T^{8} - 321187219 T^{9} - 1782148489 T^{10} - 6021692682 T^{11} - 34607406657 T^{12} + 638777810472 T^{13} + 11047710270819 T^{14} + 638777810472 p T^{15} - 34607406657 p^{2} T^{16} - 6021692682 p^{3} T^{17} - 1782148489 p^{4} T^{18} - 321187219 p^{5} T^{19} - 9914932 p^{6} T^{20} + 4853283 p^{7} T^{21} + 1000462 p^{8} T^{22} + 67958 p^{9} T^{23} - 9268 p^{10} T^{24} - 1932 p^{11} T^{25} - 41 p^{12} T^{26} + 14 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 - 10 T - 277 T^{2} + 2192 T^{3} + 49786 T^{4} - 275394 T^{5} - 6629804 T^{6} + 25018331 T^{7} + 694393718 T^{8} - 1681932119 T^{9} - 61000901305 T^{10} + 82163567860 T^{11} + 4597121803909 T^{12} - 1969709923870 T^{13} - 300133324238415 T^{14} - 1969709923870 p T^{15} + 4597121803909 p^{2} T^{16} + 82163567860 p^{3} T^{17} - 61000901305 p^{4} T^{18} - 1681932119 p^{5} T^{19} + 694393718 p^{6} T^{20} + 25018331 p^{7} T^{21} - 6629804 p^{8} T^{22} - 275394 p^{9} T^{23} + 49786 p^{10} T^{24} + 2192 p^{11} T^{25} - 277 p^{12} T^{26} - 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 - 6 T - 308 T^{2} + 164 T^{3} + 62908 T^{4} + 146213 T^{5} - 7475396 T^{6} - 42326464 T^{7} + 603793243 T^{8} + 5085070936 T^{9} - 25351846493 T^{10} - 374911832519 T^{11} + 131329441787 T^{12} + 10392590626498 T^{13} + 54642703536353 T^{14} + 10392590626498 p T^{15} + 131329441787 p^{2} T^{16} - 374911832519 p^{3} T^{17} - 25351846493 p^{4} T^{18} + 5085070936 p^{5} T^{19} + 603793243 p^{6} T^{20} - 42326464 p^{7} T^{21} - 7475396 p^{8} T^{22} + 146213 p^{9} T^{23} + 62908 p^{10} T^{24} + 164 p^{11} T^{25} - 308 p^{12} T^{26} - 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( ( 1 + T + 381 T^{2} + 417 T^{3} + 66850 T^{4} + 68550 T^{5} + 7142942 T^{6} + 6246700 T^{7} + 7142942 p T^{8} + 68550 p^{2} T^{9} + 66850 p^{3} T^{10} + 417 p^{4} T^{11} + 381 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 73 | \( 1 + 21 T - 107 T^{2} - 4532 T^{3} + 8593 T^{4} + 574123 T^{5} - 1508435 T^{6} - 52395764 T^{7} + 342478696 T^{8} + 4227610868 T^{9} - 46283154524 T^{10} - 253115011813 T^{11} + 4673586532751 T^{12} + 7744045394141 T^{13} - 369527474872486 T^{14} + 7744045394141 p T^{15} + 4673586532751 p^{2} T^{16} - 253115011813 p^{3} T^{17} - 46283154524 p^{4} T^{18} + 4227610868 p^{5} T^{19} + 342478696 p^{6} T^{20} - 52395764 p^{7} T^{21} - 1508435 p^{8} T^{22} + 574123 p^{9} T^{23} + 8593 p^{10} T^{24} - 4532 p^{11} T^{25} - 107 p^{12} T^{26} + 21 p^{13} T^{27} + p^{14} T^{28} \) | |
| 79 | \( 1 + 10 T - 226 T^{2} - 4280 T^{3} + 9610 T^{4} + 654627 T^{5} + 3201292 T^{6} - 42192356 T^{7} - 519045919 T^{8} - 901423894 T^{9} + 24953123963 T^{10} + 320449224329 T^{11} + 1694205545953 T^{12} - 14615843505140 T^{13} - 287108375549019 T^{14} - 14615843505140 p T^{15} + 1694205545953 p^{2} T^{16} + 320449224329 p^{3} T^{17} + 24953123963 p^{4} T^{18} - 901423894 p^{5} T^{19} - 519045919 p^{6} T^{20} - 42192356 p^{7} T^{21} + 3201292 p^{8} T^{22} + 654627 p^{9} T^{23} + 9610 p^{10} T^{24} - 4280 p^{11} T^{25} - 226 p^{12} T^{26} + 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 + 25 T + 157 T^{2} - 750 T^{3} - 6622 T^{4} + 69964 T^{5} + 1466905 T^{6} + 20424981 T^{7} + 74872112 T^{8} - 997051343 T^{9} + 7203133487 T^{10} + 269189728764 T^{11} + 1307717205141 T^{12} - 2467734607815 T^{13} - 39825488981322 T^{14} - 2467734607815 p T^{15} + 1307717205141 p^{2} T^{16} + 269189728764 p^{3} T^{17} + 7203133487 p^{4} T^{18} - 997051343 p^{5} T^{19} + 74872112 p^{6} T^{20} + 20424981 p^{7} T^{21} + 1466905 p^{8} T^{22} + 69964 p^{9} T^{23} - 6622 p^{10} T^{24} - 750 p^{11} T^{25} + 157 p^{12} T^{26} + 25 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 + 6 T - 395 T^{2} - 1770 T^{3} + 83413 T^{4} + 244017 T^{5} - 12791578 T^{6} - 19742595 T^{7} + 1607888462 T^{8} + 933579267 T^{9} - 173735706451 T^{10} - 11863751967 T^{11} + 16888438123377 T^{12} - 583089996186 T^{13} - 1538438146646466 T^{14} - 583089996186 p T^{15} + 16888438123377 p^{2} T^{16} - 11863751967 p^{3} T^{17} - 173735706451 p^{4} T^{18} + 933579267 p^{5} T^{19} + 1607888462 p^{6} T^{20} - 19742595 p^{7} T^{21} - 12791578 p^{8} T^{22} + 244017 p^{9} T^{23} + 83413 p^{10} T^{24} - 1770 p^{11} T^{25} - 395 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( 1 - 18 T - 332 T^{2} + 7282 T^{3} + 70792 T^{4} - 1694642 T^{5} - 11251793 T^{6} + 262778647 T^{7} + 1667779429 T^{8} - 30411590872 T^{9} - 218639545013 T^{10} + 2436064838528 T^{11} + 25890765026051 T^{12} - 94489464293929 T^{13} - 2645836565169718 T^{14} - 94489464293929 p T^{15} + 25890765026051 p^{2} T^{16} + 2436064838528 p^{3} T^{17} - 218639545013 p^{4} T^{18} - 30411590872 p^{5} T^{19} + 1667779429 p^{6} T^{20} + 262778647 p^{7} T^{21} - 11251793 p^{8} T^{22} - 1694642 p^{9} T^{23} + 70792 p^{10} T^{24} + 7282 p^{11} T^{25} - 332 p^{12} T^{26} - 18 p^{13} T^{27} + p^{14} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.23174342365138844146499163748, −2.02331241059267263527219049364, −1.99942949889810396102086488916, −1.86484586201960905448810059464, −1.78029171685700360202370871854, −1.69971141161812659102418569852, −1.68410654476034988021728328595, −1.67973503784934242998896160951, −1.60751072108291616118056880319, −1.53850107104302110541696074105, −1.38615734502827953691378107968, −1.34346846628169145087359023983, −1.34041684865990301519799361193, −1.30596482377159423276230078903, −1.17962805535861681966616271100, −0.975790000098803227993592323927, −0.968228933846238938019311347438, −0.832215919547416680191242428052, −0.63974679383569087247276314786, −0.61813576775278965711442802431, −0.59233054600610181318269403431, −0.42262515508165748128276798306, −0.17561983698752250001117077394, −0.17183232065289592471666156649, −0.11216259812435011534925736041, 0.11216259812435011534925736041, 0.17183232065289592471666156649, 0.17561983698752250001117077394, 0.42262515508165748128276798306, 0.59233054600610181318269403431, 0.61813576775278965711442802431, 0.63974679383569087247276314786, 0.832215919547416680191242428052, 0.968228933846238938019311347438, 0.975790000098803227993592323927, 1.17962805535861681966616271100, 1.30596482377159423276230078903, 1.34041684865990301519799361193, 1.34346846628169145087359023983, 1.38615734502827953691378107968, 1.53850107104302110541696074105, 1.60751072108291616118056880319, 1.67973503784934242998896160951, 1.68410654476034988021728328595, 1.69971141161812659102418569852, 1.78029171685700360202370871854, 1.86484586201960905448810059464, 1.99942949889810396102086488916, 2.02331241059267263527219049364, 2.23174342365138844146499163748
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.