Dirichlet series
| L(s) = 1 | − 5·3-s + 62·4-s + 76·7-s − 21·9-s − 310·12-s − 104·13-s + 1.61e3·16-s + 1.07e3·19-s − 380·21-s + 3.84e3·25-s − 238·27-s + 4.71e3·28-s + 3.31e3·31-s − 1.30e3·36-s − 362·37-s + 520·39-s − 6.74e3·43-s − 8.05e3·48-s − 1.03e4·49-s − 6.44e3·52-s − 5.36e3·57-s − 3.56e3·61-s − 1.59e3·63-s + 2.33e4·64-s − 1.65e4·67-s + 1.26e4·73-s − 1.92e4·75-s + ⋯ |
| L(s) = 1 | − 5/9·3-s + 31/8·4-s + 1.55·7-s − 0.259·9-s − 2.15·12-s − 0.615·13-s + 6.29·16-s + 2.96·19-s − 0.861·21-s + 6.15·25-s − 0.326·27-s + 6.01·28-s + 3.44·31-s − 1.00·36-s − 0.264·37-s + 0.341·39-s − 3.64·43-s − 3.49·48-s − 4.32·49-s − 2.38·52-s − 1.64·57-s − 0.956·61-s − 0.402·63-s + 5.71·64-s − 3.67·67-s + 2.37·73-s − 3.41·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{14}\right)^{s/2} \, \Gamma_{\C}(s+2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{14} \cdot 11^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.88879\times 10^{7}\) |
| Root analytic conductor: | \(1.84694\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 3^{14} \cdot 11^{14} ,\ ( \ : [2]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(5.079791878\) |
| \(L(\frac12)\) | \(\approx\) | \(5.079791878\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 5 T + 46 T^{2} + 191 p T^{3} + 8 p^{6} T^{4} + 865 p^{3} T^{5} + 10361 p^{4} T^{6} + 9430 p^{6} T^{7} + 10361 p^{8} T^{8} + 865 p^{11} T^{9} + 8 p^{18} T^{10} + 191 p^{17} T^{11} + 46 p^{20} T^{12} + 5 p^{24} T^{13} + p^{28} T^{14} \) |
| 11 | \( ( 1 + p^{3} T^{2} )^{7} \) | |
| good | 2 | \( 1 - 31 p T^{2} + 2233 T^{4} - 15489 p^{2} T^{6} + 189573 p^{3} T^{8} - 507249 p^{6} T^{10} + 37905449 p^{4} T^{12} - 158225893 p^{6} T^{14} + 37905449 p^{12} T^{16} - 507249 p^{22} T^{18} + 189573 p^{27} T^{20} - 15489 p^{34} T^{22} + 2233 p^{40} T^{24} - 31 p^{49} T^{26} + p^{56} T^{28} \) |
| 5 | \( 1 - 769 p T^{2} + 7248802 T^{4} - 9433228797 T^{6} + 9906146923044 T^{8} - 1769635060752627 p T^{10} + 6779542001980760729 T^{12} - \)\(45\!\cdots\!74\)\( T^{14} + 6779542001980760729 p^{8} T^{16} - 1769635060752627 p^{17} T^{18} + 9906146923044 p^{24} T^{20} - 9433228797 p^{32} T^{22} + 7248802 p^{40} T^{24} - 769 p^{49} T^{26} + p^{56} T^{28} \) | |
| 7 | \( ( 1 - 38 T + 1051 p T^{2} - 44504 p T^{3} + 32742715 T^{4} - 1355111066 T^{5} + 103333215303 T^{6} - 4090706001456 T^{7} + 103333215303 p^{4} T^{8} - 1355111066 p^{8} T^{9} + 32742715 p^{12} T^{10} - 44504 p^{17} T^{11} + 1051 p^{21} T^{12} - 38 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 13 | \( ( 1 + 4 p T + 59035 T^{2} - 3104360 T^{3} + 2755858153 T^{4} - 145996022516 T^{5} + 86746134227931 T^{6} - 8990566727995824 T^{7} + 86746134227931 p^{4} T^{8} - 145996022516 p^{8} T^{9} + 2755858153 p^{12} T^{10} - 3104360 p^{16} T^{11} + 59035 p^{20} T^{12} + 4 p^{25} T^{13} + p^{28} T^{14} )^{2} \) | |
| 17 | \( 1 - 485846 T^{2} + 129742709311 T^{4} - 25028073501120084 T^{6} + \)\(38\!\cdots\!61\)\( T^{8} - \)\(47\!\cdots\!70\)\( T^{10} + \)\(50\!\cdots\!59\)\( T^{12} - \)\(45\!\cdots\!68\)\( T^{14} + \)\(50\!\cdots\!59\)\( p^{8} T^{16} - \)\(47\!\cdots\!70\)\( p^{16} T^{18} + \)\(38\!\cdots\!61\)\( p^{24} T^{20} - 25028073501120084 p^{32} T^{22} + 129742709311 p^{40} T^{24} - 485846 p^{48} T^{26} + p^{56} T^{28} \) | |
| 19 | \( ( 1 - 536 T + 28687 p T^{2} - 216080112 T^{3} + 137229356139 T^{4} - 2328339973080 p T^{5} + 22701454646567351 T^{6} - 6335223407637576544 T^{7} + 22701454646567351 p^{4} T^{8} - 2328339973080 p^{9} T^{9} + 137229356139 p^{12} T^{10} - 216080112 p^{16} T^{11} + 28687 p^{21} T^{12} - 536 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 23 | \( 1 - 2990093 T^{2} + 4317529073506 T^{4} - 3991111390473390741 T^{6} + \)\(11\!\cdots\!08\)\( p T^{8} - \)\(13\!\cdots\!23\)\( T^{10} + \)\(52\!\cdots\!57\)\( T^{12} - \)\(16\!\cdots\!26\)\( T^{14} + \)\(52\!\cdots\!57\)\( p^{8} T^{16} - \)\(13\!\cdots\!23\)\( p^{16} T^{18} + \)\(11\!\cdots\!08\)\( p^{25} T^{20} - 3991111390473390741 p^{32} T^{22} + 4317529073506 p^{40} T^{24} - 2990093 p^{48} T^{26} + p^{56} T^{28} \) | |
| 29 | \( 1 - 5650778 T^{2} + 15779875706959 T^{4} - 28943806993315163500 T^{6} + \)\(39\!\cdots\!65\)\( T^{8} - \)\(42\!\cdots\!78\)\( T^{10} + \)\(37\!\cdots\!95\)\( T^{12} - \)\(28\!\cdots\!04\)\( T^{14} + \)\(37\!\cdots\!95\)\( p^{8} T^{16} - \)\(42\!\cdots\!78\)\( p^{16} T^{18} + \)\(39\!\cdots\!65\)\( p^{24} T^{20} - 28943806993315163500 p^{32} T^{22} + 15779875706959 p^{40} T^{24} - 5650778 p^{48} T^{26} + p^{56} T^{28} \) | |
| 31 | \( ( 1 - 1655 T + 4105918 T^{2} - 5731077719 T^{3} + 8813615578576 T^{4} - 10505503670622809 T^{5} + 11979775957570276833 T^{6} - \)\(11\!\cdots\!34\)\( T^{7} + 11979775957570276833 p^{4} T^{8} - 10505503670622809 p^{8} T^{9} + 8813615578576 p^{12} T^{10} - 5731077719 p^{16} T^{11} + 4105918 p^{20} T^{12} - 1655 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 37 | \( ( 1 + 181 T + 10236082 T^{2} + 2678247153 T^{3} + 48626479267980 T^{4} + 13313899617346179 T^{5} + \)\(13\!\cdots\!17\)\( T^{6} + \)\(33\!\cdots\!38\)\( T^{7} + \)\(13\!\cdots\!17\)\( p^{4} T^{8} + 13313899617346179 p^{8} T^{9} + 48626479267980 p^{12} T^{10} + 2678247153 p^{16} T^{11} + 10236082 p^{20} T^{12} + 181 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 41 | \( 1 - 20067002 T^{2} + 193946959226095 T^{4} - \)\(11\!\cdots\!60\)\( T^{6} + \)\(52\!\cdots\!85\)\( T^{8} - \)\(18\!\cdots\!34\)\( T^{10} + \)\(53\!\cdots\!31\)\( T^{12} - \)\(14\!\cdots\!76\)\( T^{14} + \)\(53\!\cdots\!31\)\( p^{8} T^{16} - \)\(18\!\cdots\!34\)\( p^{16} T^{18} + \)\(52\!\cdots\!85\)\( p^{24} T^{20} - \)\(11\!\cdots\!60\)\( p^{32} T^{22} + 193946959226095 p^{40} T^{24} - 20067002 p^{48} T^{26} + p^{56} T^{28} \) | |
| 43 | \( ( 1 + 3370 T + 17487001 T^{2} + 958502448 p T^{3} + 116672249476671 T^{4} + 222115030094308806 T^{5} + \)\(47\!\cdots\!39\)\( T^{6} + \)\(82\!\cdots\!12\)\( T^{7} + \)\(47\!\cdots\!39\)\( p^{4} T^{8} + 222115030094308806 p^{8} T^{9} + 116672249476671 p^{12} T^{10} + 958502448 p^{17} T^{11} + 17487001 p^{20} T^{12} + 3370 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 47 | \( 1 - 28348958 T^{2} + 434446812146299 T^{4} - \)\(47\!\cdots\!88\)\( T^{6} + \)\(41\!\cdots\!25\)\( T^{8} - \)\(29\!\cdots\!10\)\( T^{10} + \)\(17\!\cdots\!35\)\( T^{12} - \)\(94\!\cdots\!84\)\( T^{14} + \)\(17\!\cdots\!35\)\( p^{8} T^{16} - \)\(29\!\cdots\!10\)\( p^{16} T^{18} + \)\(41\!\cdots\!25\)\( p^{24} T^{20} - \)\(47\!\cdots\!88\)\( p^{32} T^{22} + 434446812146299 p^{40} T^{24} - 28348958 p^{48} T^{26} + p^{56} T^{28} \) | |
| 53 | \( 1 - 68012342 T^{2} + 2294298247507435 T^{4} - \)\(50\!\cdots\!52\)\( T^{6} + \)\(83\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!18\)\( T^{10} + \)\(11\!\cdots\!95\)\( T^{12} - \)\(97\!\cdots\!24\)\( T^{14} + \)\(11\!\cdots\!95\)\( p^{8} T^{16} - \)\(10\!\cdots\!18\)\( p^{16} T^{18} + \)\(83\!\cdots\!05\)\( p^{24} T^{20} - \)\(50\!\cdots\!52\)\( p^{32} T^{22} + 2294298247507435 p^{40} T^{24} - 68012342 p^{48} T^{26} + p^{56} T^{28} \) | |
| 59 | \( 1 - 94058909 T^{2} + 72262552587182 p T^{4} - \)\(12\!\cdots\!69\)\( T^{6} + \)\(26\!\cdots\!12\)\( T^{8} - \)\(43\!\cdots\!23\)\( T^{10} + \)\(60\!\cdots\!13\)\( T^{12} - \)\(76\!\cdots\!30\)\( T^{14} + \)\(60\!\cdots\!13\)\( p^{8} T^{16} - \)\(43\!\cdots\!23\)\( p^{16} T^{18} + \)\(26\!\cdots\!12\)\( p^{24} T^{20} - \)\(12\!\cdots\!69\)\( p^{32} T^{22} + 72262552587182 p^{41} T^{24} - 94058909 p^{48} T^{26} + p^{56} T^{28} \) | |
| 61 | \( ( 1 + 1780 T + 41912563 T^{2} + 97658312472 T^{3} + 1128391765815297 T^{4} + 2332428541262115660 T^{5} + \)\(21\!\cdots\!39\)\( T^{6} + \)\(39\!\cdots\!20\)\( T^{7} + \)\(21\!\cdots\!39\)\( p^{4} T^{8} + 2332428541262115660 p^{8} T^{9} + 1128391765815297 p^{12} T^{10} + 97658312472 p^{16} T^{11} + 41912563 p^{20} T^{12} + 1780 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 67 | \( ( 1 + 8257 T + 78448894 T^{2} + 418710202913 T^{3} + 2726921864628272 T^{4} + 11796880355207077727 T^{5} + \)\(64\!\cdots\!45\)\( T^{6} + \)\(25\!\cdots\!86\)\( T^{7} + \)\(64\!\cdots\!45\)\( p^{4} T^{8} + 11796880355207077727 p^{8} T^{9} + 2726921864628272 p^{12} T^{10} + 418710202913 p^{16} T^{11} + 78448894 p^{20} T^{12} + 8257 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 71 | \( 1 - 203956901 T^{2} + 269209017678086 p T^{4} - \)\(10\!\cdots\!93\)\( T^{6} + \)\(40\!\cdots\!72\)\( T^{8} - \)\(10\!\cdots\!75\)\( T^{10} + \)\(19\!\cdots\!77\)\( T^{12} - \)\(39\!\cdots\!18\)\( T^{14} + \)\(19\!\cdots\!77\)\( p^{8} T^{16} - \)\(10\!\cdots\!75\)\( p^{16} T^{18} + \)\(40\!\cdots\!72\)\( p^{24} T^{20} - \)\(10\!\cdots\!93\)\( p^{32} T^{22} + 269209017678086 p^{41} T^{24} - 203956901 p^{48} T^{26} + p^{56} T^{28} \) | |
| 73 | \( ( 1 - 6332 T + 112811287 T^{2} - 949500906536 T^{3} + 7258163516936341 T^{4} - 54948310613542514372 T^{5} + \)\(33\!\cdots\!35\)\( T^{6} - \)\(18\!\cdots\!16\)\( T^{7} + \)\(33\!\cdots\!35\)\( p^{4} T^{8} - 54948310613542514372 p^{8} T^{9} + 7258163516936341 p^{12} T^{10} - 949500906536 p^{16} T^{11} + 112811287 p^{20} T^{12} - 6332 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 79 | \( ( 1 - 1526 T + 204404437 T^{2} - 322950507640 T^{3} + 19831551969918467 T^{4} - 29336944829432468842 T^{5} + \)\(11\!\cdots\!79\)\( T^{6} - \)\(14\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!79\)\( p^{4} T^{8} - 29336944829432468842 p^{8} T^{9} + 19831551969918467 p^{12} T^{10} - 322950507640 p^{16} T^{11} + 204404437 p^{20} T^{12} - 1526 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 83 | \( 1 - 343468394 T^{2} + 60970942911104059 T^{4} - \)\(73\!\cdots\!96\)\( T^{6} + \)\(67\!\cdots\!53\)\( T^{8} - \)\(50\!\cdots\!62\)\( T^{10} + \)\(30\!\cdots\!83\)\( T^{12} - \)\(15\!\cdots\!72\)\( T^{14} + \)\(30\!\cdots\!83\)\( p^{8} T^{16} - \)\(50\!\cdots\!62\)\( p^{16} T^{18} + \)\(67\!\cdots\!53\)\( p^{24} T^{20} - \)\(73\!\cdots\!96\)\( p^{32} T^{22} + 60970942911104059 p^{40} T^{24} - 343468394 p^{48} T^{26} + p^{56} T^{28} \) | |
| 89 | \( 1 - 392958917 T^{2} + 74853447914451754 T^{4} - \)\(92\!\cdots\!25\)\( T^{6} + \)\(84\!\cdots\!84\)\( T^{8} - \)\(63\!\cdots\!11\)\( T^{10} + \)\(41\!\cdots\!65\)\( T^{12} - \)\(26\!\cdots\!18\)\( T^{14} + \)\(41\!\cdots\!65\)\( p^{8} T^{16} - \)\(63\!\cdots\!11\)\( p^{16} T^{18} + \)\(84\!\cdots\!84\)\( p^{24} T^{20} - \)\(92\!\cdots\!25\)\( p^{32} T^{22} + 74853447914451754 p^{40} T^{24} - 392958917 p^{48} T^{26} + p^{56} T^{28} \) | |
| 97 | \( ( 1 + 13927 T + 376447066 T^{2} + 3249505605443 T^{3} + 55136591519145860 T^{4} + \)\(32\!\cdots\!61\)\( T^{5} + \)\(52\!\cdots\!65\)\( T^{6} + \)\(26\!\cdots\!26\)\( T^{7} + \)\(52\!\cdots\!65\)\( p^{4} T^{8} + \)\(32\!\cdots\!61\)\( p^{8} T^{9} + 55136591519145860 p^{12} T^{10} + 3249505605443 p^{16} T^{11} + 376447066 p^{20} T^{12} + 13927 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
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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
−4.95715528684772585448685479544, −4.92995144849624875494967281955, −4.88486486135562505300935594334, −4.85517156590110675675082396981, −4.63791400446325096045324781152, −4.60762930465750402776610778399, −4.48766759596913104322385669418, −4.01121938847311481563989521074, −3.54749921223883744287510051736, −3.53631342676135792942963530382, −3.49339954739542190199337775099, −3.20043040559172205757490383512, −3.14130691068360885392875755281, −2.88375208383835702445690962722, −2.69006146467603872202887313513, −2.67953471915558146886243651448, −2.52729682595026094871170167433, −2.32120308897522498957968358141, −1.95668260461259627631196430166, −1.59853907764022310733962055026, −1.40843251256726647481537590499, −1.27901584887210153739637928356, −1.19492086520459920349295247127, −0.954783677693226214468085483630, −0.13576013610252794419969532596, 0.13576013610252794419969532596, 0.954783677693226214468085483630, 1.19492086520459920349295247127, 1.27901584887210153739637928356, 1.40843251256726647481537590499, 1.59853907764022310733962055026, 1.95668260461259627631196430166, 2.32120308897522498957968358141, 2.52729682595026094871170167433, 2.67953471915558146886243651448, 2.69006146467603872202887313513, 2.88375208383835702445690962722, 3.14130691068360885392875755281, 3.20043040559172205757490383512, 3.49339954739542190199337775099, 3.53631342676135792942963530382, 3.54749921223883744287510051736, 4.01121938847311481563989521074, 4.48766759596913104322385669418, 4.60762930465750402776610778399, 4.63791400446325096045324781152, 4.85517156590110675675082396981, 4.88486486135562505300935594334, 4.92995144849624875494967281955, 4.95715528684772585448685479544
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.