Dirichlet series
| L(s) = 1 | + 3·3-s − 2·5-s + 6·7-s + 2·9-s + 2·11-s + 2·13-s − 6·15-s + 2·17-s + 7·19-s + 18·21-s + 11·23-s + 15·25-s + 29-s + 2·31-s + 6·33-s − 12·35-s + 10·37-s + 6·39-s − 33·41-s + 7·43-s − 4·45-s + 6·47-s + 16·49-s + 6·51-s − 15·53-s − 4·55-s + 21·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s + 2.26·7-s + 2/3·9-s + 0.603·11-s + 0.554·13-s − 1.54·15-s + 0.485·17-s + 1.60·19-s + 3.92·21-s + 2.29·23-s + 3·25-s + 0.185·29-s + 0.359·31-s + 1.04·33-s − 2.02·35-s + 1.64·37-s + 0.960·39-s − 5.15·41-s + 1.06·43-s − 0.596·45-s + 0.875·47-s + 16/7·49-s + 0.840·51-s − 2.06·53-s − 0.539·55-s + 2.78·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{28} \cdot 7^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{28} \cdot 7^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{28} \cdot 3^{28} \cdot 7^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(17843.7\) |
| Root analytic conductor: | \(1.41853\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{28} \cdot 3^{28} \cdot 7^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(16.12569282\) |
| \(L(\frac12)\) | \(\approx\) | \(16.12569282\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - p T + 7 T^{2} - 5 p T^{3} + 34 T^{4} - 26 p T^{5} + 16 p^{2} T^{6} - 28 p^{2} T^{7} + 16 p^{3} T^{8} - 26 p^{3} T^{9} + 34 p^{3} T^{10} - 5 p^{5} T^{11} + 7 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 6 T + 20 T^{2} - 11 p T^{3} + 309 T^{4} - 961 T^{5} + 2706 T^{6} - 7572 T^{7} + 2706 p T^{8} - 961 p^{2} T^{9} + 309 p^{3} T^{10} - 11 p^{5} T^{11} + 20 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| good | 5 | \( 1 + 2 T - 11 T^{2} - 66 T^{3} - p^{2} T^{4} + 533 T^{5} + 308 p T^{6} - 603 T^{7} - 11032 T^{8} - 21691 T^{9} + 797 p^{2} T^{10} + 31389 p T^{11} + 46761 p T^{12} - 385788 T^{13} - 1886454 T^{14} - 385788 p T^{15} + 46761 p^{3} T^{16} + 31389 p^{4} T^{17} + 797 p^{6} T^{18} - 21691 p^{5} T^{19} - 11032 p^{6} T^{20} - 603 p^{7} T^{21} + 308 p^{9} T^{22} + 533 p^{9} T^{23} - p^{12} T^{24} - 66 p^{11} T^{25} - 11 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) |
| 11 | \( 1 - 2 T - 32 T^{2} + 42 T^{3} + 722 T^{4} - 614 T^{5} - 9113 T^{6} - 2733 T^{7} + 72959 T^{8} + 126076 T^{9} + 256457 T^{10} - 2291538 T^{11} - 13152543 T^{12} + 8654619 T^{13} + 213521742 T^{14} + 8654619 p T^{15} - 13152543 p^{2} T^{16} - 2291538 p^{3} T^{17} + 256457 p^{4} T^{18} + 126076 p^{5} T^{19} + 72959 p^{6} T^{20} - 2733 p^{7} T^{21} - 9113 p^{8} T^{22} - 614 p^{9} T^{23} + 722 p^{10} T^{24} + 42 p^{11} T^{25} - 32 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 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 | \( 1 - 11 T - 8 T^{2} + 333 T^{3} + 401 T^{4} - 3407 T^{5} - 42005 T^{6} + 36987 T^{7} + 1121822 T^{8} + 103015 T^{9} - 29774440 T^{10} + 52947609 T^{11} + 527481093 T^{12} - 1421356596 T^{13} - 4817982924 T^{14} - 1421356596 p T^{15} + 527481093 p^{2} T^{16} + 52947609 p^{3} T^{17} - 29774440 p^{4} T^{18} + 103015 p^{5} T^{19} + 1121822 p^{6} T^{20} + 36987 p^{7} T^{21} - 42005 p^{8} T^{22} - 3407 p^{9} T^{23} + 401 p^{10} T^{24} + 333 p^{11} T^{25} - 8 p^{12} T^{26} - 11 p^{13} T^{27} + p^{14} T^{28} \) | |
| 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 | \( ( 1 - T + 86 T^{2} - 194 T^{3} + 4879 T^{4} - 9670 T^{5} + 191454 T^{6} - 403758 T^{7} + 191454 p T^{8} - 9670 p^{2} T^{9} + 4879 p^{3} T^{10} - 194 p^{4} T^{11} + 86 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 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 | \( ( 1 - 3 T + 224 T^{2} - 711 T^{3} + 24846 T^{4} - 72810 T^{5} + 1742359 T^{6} - 4337544 T^{7} + 1742359 p T^{8} - 72810 p^{2} T^{9} + 24846 p^{3} T^{10} - 711 p^{4} T^{11} + 224 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 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 | \( ( 1 - 14 T + 237 T^{2} - 2625 T^{3} + 28687 T^{4} - 249069 T^{5} + 2197811 T^{6} - 16839260 T^{7} + 2197811 p T^{8} - 249069 p^{2} T^{9} + 28687 p^{3} T^{10} - 2625 p^{4} T^{11} + 237 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 61 | \( ( 1 - 10 T + 377 T^{2} - 2981 T^{3} + 62533 T^{4} - 402217 T^{5} + 6040473 T^{6} - 31437684 T^{7} + 6040473 p T^{8} - 402217 p^{2} T^{9} + 62533 p^{3} T^{10} - 2981 p^{4} T^{11} + 377 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 67 | \( ( 1 + 6 T + 344 T^{2} + 1114 T^{3} + 48744 T^{4} + 55499 T^{5} + 4192269 T^{6} + 1055346 T^{7} + 4192269 p T^{8} + 55499 p^{2} T^{9} + 48744 p^{3} T^{10} + 1114 p^{4} T^{11} + 344 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 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 | \( ( 1 - 10 T + 326 T^{2} - 3770 T^{3} + 58966 T^{4} - 604927 T^{5} + 7130289 T^{6} - 58615338 T^{7} + 7130289 p T^{8} - 604927 p^{2} T^{9} + 58966 p^{3} T^{10} - 3770 p^{4} T^{11} + 326 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 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
−3.63306224985434148952944158498, −3.57092669392556173013210634043, −3.54185640376500686899125784813, −3.52973001635413369069259241409, −3.42795247126110740925294418258, −3.39761808121201185183701185234, −3.06106403327273518282555859299, −2.97097173802909780009815355855, −2.90992512117847082291694465823, −2.83420961613768087635905577744, −2.78680738046876113655481131169, −2.65448382001076245664708238447, −2.62592708152449770135714455476, −2.25159853448712322578282531929, −2.18069343014687997377646021356, −2.17092269136658864600564506600, −2.09294840209048169578169593544, −1.70353336615132151342990655443, −1.68838356305210669205885573198, −1.47866794895013501675257818653, −1.26849478989146811271247865616, −1.10161820228680079339764307731, −1.04619314307125930409596184284, −0.916857380986240354336948022301, −0.61055075501213933325141314681, 0.61055075501213933325141314681, 0.916857380986240354336948022301, 1.04619314307125930409596184284, 1.10161820228680079339764307731, 1.26849478989146811271247865616, 1.47866794895013501675257818653, 1.68838356305210669205885573198, 1.70353336615132151342990655443, 2.09294840209048169578169593544, 2.17092269136658864600564506600, 2.18069343014687997377646021356, 2.25159853448712322578282531929, 2.62592708152449770135714455476, 2.65448382001076245664708238447, 2.78680738046876113655481131169, 2.83420961613768087635905577744, 2.90992512117847082291694465823, 2.97097173802909780009815355855, 3.06106403327273518282555859299, 3.39761808121201185183701185234, 3.42795247126110740925294418258, 3.52973001635413369069259241409, 3.54185640376500686899125784813, 3.57092669392556173013210634043, 3.63306224985434148952944158498
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.