Dirichlet series
| L(s) = 1 | − 342·2-s − 2.95e4·3-s − 7.01e5·4-s − 2.48e6·5-s + 1.00e7·6-s + 1.61e8·7-s + 4.49e8·8-s − 2.05e9·9-s + 8.50e8·10-s − 5.23e9·11-s + 2.07e10·12-s − 2.40e10·13-s − 5.52e10·14-s + 7.34e10·15-s − 7.10e10·16-s − 1.12e12·17-s + 7.03e11·18-s − 1.68e12·19-s + 1.74e12·20-s − 4.76e12·21-s + 1.78e12·22-s − 2.03e13·23-s − 1.32e13·24-s − 4.11e13·25-s + 8.23e12·26-s + 8.45e13·27-s − 1.13e14·28-s + ⋯ |
| L(s) = 1 | − 0.472·2-s − 0.866·3-s − 1.33·4-s − 0.569·5-s + 0.409·6-s + 1.51·7-s + 1.18·8-s − 1.76·9-s + 0.268·10-s − 0.669·11-s + 1.15·12-s − 0.629·13-s − 0.714·14-s + 0.493·15-s − 0.258·16-s − 2.29·17-s + 0.835·18-s − 1.20·19-s + 0.762·20-s − 1.30·21-s + 0.316·22-s − 2.35·23-s − 1.02·24-s − 2.15·25-s + 0.297·26-s + 2.13·27-s − 2.02·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2401\) = \(7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(65817.7\) |
| Root analytic conductor: | \(4.00214\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2401,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\) |
Particular Values
| \(L(10)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{21}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 7 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 171 p T + 51185 p^{4} T^{2} + 276237 p^{8} T^{3} + 63043611 p^{13} T^{4} + 276237 p^{27} T^{5} + 51185 p^{42} T^{6} + 171 p^{58} T^{7} + p^{76} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 9842 p T + 108446944 p^{3} T^{2} + 257750599274 p^{5} T^{3} + 607570963534318 p^{8} T^{4} + 257750599274 p^{24} T^{5} + 108446944 p^{41} T^{6} + 9842 p^{58} T^{7} + p^{76} T^{8} \) | |
| 5 | $C_2 \wr S_4$ | \( 1 + 497322 p T + 1892477739092 p^{2} T^{2} + 649249530552821142 p^{3} T^{3} + \)\(19\!\cdots\!86\)\( p^{4} T^{4} + 649249530552821142 p^{22} T^{5} + 1892477739092 p^{40} T^{6} + 497322 p^{58} T^{7} + p^{76} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 5232894012 T + \)\(16\!\cdots\!48\)\( T^{2} + \)\(84\!\cdots\!24\)\( p T^{3} + \)\(10\!\cdots\!50\)\( p^{2} T^{4} + \)\(84\!\cdots\!24\)\( p^{20} T^{5} + \)\(16\!\cdots\!48\)\( p^{38} T^{6} + 5232894012 p^{57} T^{7} + p^{76} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 24071694934 T + \)\(27\!\cdots\!00\)\( T^{2} + \)\(56\!\cdots\!22\)\( p T^{3} + \)\(31\!\cdots\!38\)\( p^{2} T^{4} + \)\(56\!\cdots\!22\)\( p^{20} T^{5} + \)\(27\!\cdots\!00\)\( p^{38} T^{6} + 24071694934 p^{57} T^{7} + p^{76} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 1122693554556 T + \)\(80\!\cdots\!04\)\( p T^{2} + \)\(29\!\cdots\!80\)\( p^{2} T^{3} + \)\(11\!\cdots\!22\)\( p^{3} T^{4} + \)\(29\!\cdots\!80\)\( p^{21} T^{5} + \)\(80\!\cdots\!04\)\( p^{39} T^{6} + 1122693554556 p^{57} T^{7} + p^{76} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 88896545878 p T + \)\(65\!\cdots\!36\)\( T^{2} + \)\(46\!\cdots\!46\)\( p T^{3} + \)\(18\!\cdots\!06\)\( T^{4} + \)\(46\!\cdots\!46\)\( p^{20} T^{5} + \)\(65\!\cdots\!36\)\( p^{38} T^{6} + 88896545878 p^{58} T^{7} + p^{76} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 884490020424 p T + \)\(20\!\cdots\!32\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(59\!\cdots\!46\)\( T^{4} + \)\(10\!\cdots\!80\)\( p^{19} T^{5} + \)\(20\!\cdots\!32\)\( p^{38} T^{6} + 884490020424 p^{58} T^{7} + p^{76} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 17794845083772 T + \)\(25\!\cdots\!76\)\( T^{2} - \)\(57\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(57\!\cdots\!04\)\( p^{19} T^{5} + \)\(25\!\cdots\!76\)\( p^{38} T^{6} - 17794845083772 p^{57} T^{7} + p^{76} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 438619343652812 T + \)\(13\!\cdots\!12\)\( T^{2} - \)\(27\!\cdots\!36\)\( T^{3} + \)\(47\!\cdots\!50\)\( T^{4} - \)\(27\!\cdots\!36\)\( p^{19} T^{5} + \)\(13\!\cdots\!12\)\( p^{38} T^{6} - 438619343652812 p^{57} T^{7} + p^{76} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 371101054682492 T + \)\(10\!\cdots\!72\)\( T^{2} - \)\(25\!\cdots\!36\)\( T^{3} + \)\(79\!\cdots\!78\)\( T^{4} - \)\(25\!\cdots\!36\)\( p^{19} T^{5} + \)\(10\!\cdots\!72\)\( p^{38} T^{6} - 371101054682492 p^{57} T^{7} + p^{76} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 3519649536072516 T + \)\(12\!\cdots\!84\)\( T^{2} + \)\(32\!\cdots\!64\)\( T^{3} + \)\(73\!\cdots\!30\)\( T^{4} + \)\(32\!\cdots\!64\)\( p^{19} T^{5} + \)\(12\!\cdots\!84\)\( p^{38} T^{6} + 3519649536072516 p^{57} T^{7} + p^{76} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 4463367374178964 T + \)\(41\!\cdots\!40\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{3} + \)\(67\!\cdots\!02\)\( T^{4} + \)\(13\!\cdots\!36\)\( p^{19} T^{5} + \)\(41\!\cdots\!40\)\( p^{38} T^{6} + 4463367374178964 p^{57} T^{7} + p^{76} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 12676380373106604 T + \)\(10\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!54\)\( T^{4} + \)\(11\!\cdots\!76\)\( p^{19} T^{5} + \)\(10\!\cdots\!84\)\( p^{38} T^{6} + 12676380373106604 p^{57} T^{7} + p^{76} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 15595612222940712 T + \)\(12\!\cdots\!16\)\( T^{2} - \)\(84\!\cdots\!12\)\( T^{3} + \)\(78\!\cdots\!54\)\( T^{4} - \)\(84\!\cdots\!12\)\( p^{19} T^{5} + \)\(12\!\cdots\!16\)\( p^{38} T^{6} - 15595612222940712 p^{57} T^{7} + p^{76} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 128722843456786530 T + \)\(14\!\cdots\!76\)\( T^{2} + \)\(13\!\cdots\!10\)\( T^{3} + \)\(97\!\cdots\!86\)\( T^{4} + \)\(13\!\cdots\!10\)\( p^{19} T^{5} + \)\(14\!\cdots\!76\)\( p^{38} T^{6} + 128722843456786530 p^{57} T^{7} + p^{76} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 99854565167649586 T + \)\(15\!\cdots\!84\)\( T^{2} + \)\(15\!\cdots\!14\)\( T^{3} + \)\(19\!\cdots\!70\)\( T^{4} + \)\(15\!\cdots\!14\)\( p^{19} T^{5} + \)\(15\!\cdots\!84\)\( p^{38} T^{6} + 99854565167649586 p^{57} T^{7} + p^{76} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 559752529353715960 T + \)\(28\!\cdots\!56\)\( T^{2} + \)\(84\!\cdots\!48\)\( T^{3} + \)\(22\!\cdots\!30\)\( T^{4} + \)\(84\!\cdots\!48\)\( p^{19} T^{5} + \)\(28\!\cdots\!56\)\( p^{38} T^{6} + 559752529353715960 p^{57} T^{7} + p^{76} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 431823453358672584 T + \)\(37\!\cdots\!36\)\( T^{2} + \)\(12\!\cdots\!84\)\( p T^{3} + \)\(62\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!84\)\( p^{20} T^{5} + \)\(37\!\cdots\!36\)\( p^{38} T^{6} + 431823453358672584 p^{57} T^{7} + p^{76} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 141778013178404848 T + \)\(17\!\cdots\!96\)\( T^{2} - \)\(17\!\cdots\!52\)\( T^{3} + \)\(87\!\cdots\!34\)\( T^{4} - \)\(17\!\cdots\!52\)\( p^{19} T^{5} + \)\(17\!\cdots\!96\)\( p^{38} T^{6} + 141778013178404848 p^{57} T^{7} + p^{76} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 278647331872313704 T + \)\(81\!\cdots\!16\)\( T^{2} - \)\(70\!\cdots\!72\)\( T^{3} - \)\(84\!\cdots\!14\)\( T^{4} - \)\(70\!\cdots\!72\)\( p^{19} T^{5} + \)\(81\!\cdots\!16\)\( p^{38} T^{6} + 278647331872313704 p^{57} T^{7} + p^{76} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 2246245523526488778 T + \)\(52\!\cdots\!16\)\( T^{2} + \)\(92\!\cdots\!98\)\( T^{3} + \)\(23\!\cdots\!34\)\( T^{4} + \)\(92\!\cdots\!98\)\( p^{19} T^{5} + \)\(52\!\cdots\!16\)\( p^{38} T^{6} + 2246245523526488778 p^{57} T^{7} + p^{76} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 7106866925026829688 T + \)\(53\!\cdots\!76\)\( T^{2} + \)\(23\!\cdots\!16\)\( T^{3} + \)\(93\!\cdots\!06\)\( T^{4} + \)\(23\!\cdots\!16\)\( p^{19} T^{5} + \)\(53\!\cdots\!76\)\( p^{38} T^{6} + 7106866925026829688 p^{57} T^{7} + p^{76} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 237692331127149724 T + \)\(83\!\cdots\!84\)\( T^{2} - \)\(24\!\cdots\!44\)\( T^{3} + \)\(75\!\cdots\!34\)\( T^{4} - \)\(24\!\cdots\!44\)\( p^{19} T^{5} + \)\(83\!\cdots\!84\)\( p^{38} T^{6} + 237692331127149724 p^{57} T^{7} + p^{76} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.34995167705682554054475642885, −12.22477876007795019680136049185, −12.05291486131442498689500381760, −11.59137280575009835117694139473, −11.36634271262849477444521619756, −11.08742661382310118009624294745, −10.37616371830670322390208434067, −10.06930587477902549280815100670, −9.651065259036058371027794731018, −8.620806180965386741682291782578, −8.567619264575169064616009084134, −8.451604277298710259591383598941, −7.942816051957381269279983997265, −7.48109291789986106217199980312, −6.54793043199558562728409887897, −6.03884131910157386436984850110, −5.85207065300472202729559822752, −4.82023176556919691902227622910, −4.65840038934072170891488764601, −4.54614521439484996501420159174, −3.99792023494363782959433374071, −2.95481638518228861201608468411, −2.34690008206409307427150395869, −1.93941704330094800383367190337, −1.35433081639575334634772592864, 0, 0, 0, 0, 1.35433081639575334634772592864, 1.93941704330094800383367190337, 2.34690008206409307427150395869, 2.95481638518228861201608468411, 3.99792023494363782959433374071, 4.54614521439484996501420159174, 4.65840038934072170891488764601, 4.82023176556919691902227622910, 5.85207065300472202729559822752, 6.03884131910157386436984850110, 6.54793043199558562728409887897, 7.48109291789986106217199980312, 7.942816051957381269279983997265, 8.451604277298710259591383598941, 8.567619264575169064616009084134, 8.620806180965386741682291782578, 9.651065259036058371027794731018, 10.06930587477902549280815100670, 10.37616371830670322390208434067, 11.08742661382310118009624294745, 11.36634271262849477444521619756, 11.59137280575009835117694139473, 12.05291486131442498689500381760, 12.22477876007795019680136049185, 13.34995167705682554054475642885