Dirichlet series
| L(s) = 1 | − 15·2-s − 7.85e3·4-s + 6.25e4·5-s + 3.43e5·7-s + 1.54e5·8-s − 9.37e5·10-s + 1.26e7·11-s + 3.43e7·13-s − 5.14e6·14-s + 1.24e7·16-s + 8.43e7·17-s − 1.31e8·19-s − 4.90e8·20-s − 1.90e8·22-s + 1.04e9·23-s + 2.44e9·25-s − 5.15e8·26-s − 2.69e9·28-s − 1.82e9·29-s + 3.50e9·31-s − 7.17e8·32-s − 1.26e9·34-s + 2.14e10·35-s + 8.29e9·37-s + 1.97e9·38-s + 9.68e9·40-s + 5.44e10·41-s + ⋯ |
| L(s) = 1 | − 0.165·2-s − 0.958·4-s + 1.78·5-s + 1.10·7-s + 0.208·8-s − 0.296·10-s + 2.16·11-s + 1.97·13-s − 0.182·14-s + 0.185·16-s + 0.847·17-s − 0.642·19-s − 1.71·20-s − 0.358·22-s + 1.47·23-s + 2·25-s − 0.326·26-s − 1.05·28-s − 0.570·29-s + 0.708·31-s − 0.118·32-s − 0.140·34-s + 1.97·35-s + 0.531·37-s + 0.106·38-s + 0.373·40-s + 1.79·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(4100625\) = \(3^{8} \cdot 5^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.42163\times 10^{6}\) |
| Root analytic conductor: | \(6.94650\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 4100625,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\) |
Particular Values
| \(L(7)\) | \(\approx\) | \(23.05851113\) |
| \(L(\frac12)\) | \(\approx\) | \(23.05851113\) |
| \(L(\frac{15}{2})\) | 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 | 3 | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 15 T + 4039 p T^{2} + 10505 p^{3} T^{3} + 390033 p^{7} T^{4} + 10505 p^{16} T^{5} + 4039 p^{27} T^{6} + 15 p^{39} T^{7} + p^{52} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 343040 T + 194313329428 T^{2} - 11990142535557120 p T^{3} + \)\(52\!\cdots\!06\)\( p^{2} T^{4} - 11990142535557120 p^{14} T^{5} + 194313329428 p^{26} T^{6} - 343040 p^{39} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 12697800 T + 14778711137884 p T^{2} - 9298384593225367400 p^{2} T^{3} + \)\(62\!\cdots\!86\)\( p^{3} T^{4} - 9298384593225367400 p^{15} T^{5} + 14778711137884 p^{27} T^{6} - 12697800 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 34336040 T + 117482613371524 p T^{2} - \)\(31\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!54\)\( T^{4} - \)\(31\!\cdots\!60\)\( p^{13} T^{5} + 117482613371524 p^{27} T^{6} - 34336040 p^{39} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 84377280 T + 15521353071831788 T^{2} - \)\(64\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!74\)\( T^{4} - \)\(64\!\cdots\!80\)\( p^{13} T^{5} + 15521353071831788 p^{26} T^{6} - 84377280 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 131821144 T + 7714368091215748 p T^{2} + \)\(15\!\cdots\!12\)\( T^{3} + \)\(87\!\cdots\!70\)\( T^{4} + \)\(15\!\cdots\!12\)\( p^{13} T^{5} + 7714368091215748 p^{27} T^{6} + 131821144 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 1046106360 T + 1818308015374219292 T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!94\)\( T^{4} - \)\(15\!\cdots\!40\)\( p^{13} T^{5} + 1818308015374219292 p^{26} T^{6} - 1046106360 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 1826812200 T + 31045034855939699756 T^{2} + \)\(48\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!26\)\( T^{4} + \)\(48\!\cdots\!00\)\( p^{13} T^{5} + 31045034855939699756 p^{26} T^{6} + 1826812200 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 3501222872 T + 60142659648538757308 T^{2} - \)\(61\!\cdots\!64\)\( p T^{3} + \)\(21\!\cdots\!70\)\( T^{4} - \)\(61\!\cdots\!64\)\( p^{14} T^{5} + 60142659648538757308 p^{26} T^{6} - 3501222872 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 8297642120 T + \)\(65\!\cdots\!88\)\( T^{2} - \)\(58\!\cdots\!20\)\( T^{3} + \)\(21\!\cdots\!54\)\( T^{4} - \)\(58\!\cdots\!20\)\( p^{13} T^{5} + \)\(65\!\cdots\!88\)\( p^{26} T^{6} - 8297642120 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 54459122400 T + \)\(22\!\cdots\!84\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!46\)\( T^{4} - \)\(10\!\cdots\!00\)\( p^{13} T^{5} + \)\(22\!\cdots\!84\)\( p^{26} T^{6} - 54459122400 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 48173425360 T + \)\(45\!\cdots\!72\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!94\)\( T^{4} + \)\(21\!\cdots\!40\)\( p^{13} T^{5} + \)\(45\!\cdots\!72\)\( p^{26} T^{6} + 48173425360 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 202496682840 T + \)\(31\!\cdots\!68\)\( T^{2} - \)\(31\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!14\)\( T^{4} - \)\(31\!\cdots\!40\)\( p^{13} T^{5} + \)\(31\!\cdots\!68\)\( p^{26} T^{6} - 202496682840 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 116147122920 T + \)\(62\!\cdots\!32\)\( T^{2} - \)\(81\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!14\)\( T^{4} - \)\(81\!\cdots\!80\)\( p^{13} T^{5} + \)\(62\!\cdots\!32\)\( p^{26} T^{6} - 116147122920 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 885100702200 T + \)\(55\!\cdots\!16\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(92\!\cdots\!46\)\( T^{4} - \)\(25\!\cdots\!00\)\( p^{13} T^{5} + \)\(55\!\cdots\!16\)\( p^{26} T^{6} - 885100702200 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 467836185368 T + \)\(31\!\cdots\!08\)\( T^{2} - \)\(11\!\cdots\!76\)\( T^{3} + \)\(41\!\cdots\!70\)\( T^{4} - \)\(11\!\cdots\!76\)\( p^{13} T^{5} + \)\(31\!\cdots\!08\)\( p^{26} T^{6} - 467836185368 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 819037749440 T + \)\(93\!\cdots\!48\)\( T^{2} - \)\(70\!\cdots\!40\)\( T^{3} + \)\(85\!\cdots\!14\)\( T^{4} - \)\(70\!\cdots\!40\)\( p^{13} T^{5} + \)\(93\!\cdots\!48\)\( p^{26} T^{6} - 819037749440 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 2886468559200 T + \)\(41\!\cdots\!44\)\( T^{2} - \)\(52\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!26\)\( T^{4} - \)\(52\!\cdots\!00\)\( p^{13} T^{5} + \)\(41\!\cdots\!44\)\( p^{26} T^{6} - 2886468559200 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1235535876280 T + \)\(37\!\cdots\!32\)\( T^{2} + \)\(56\!\cdots\!20\)\( T^{3} + \)\(68\!\cdots\!34\)\( T^{4} + \)\(56\!\cdots\!20\)\( p^{13} T^{5} + \)\(37\!\cdots\!32\)\( p^{26} T^{6} + 1235535876280 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 3553486866616 T + \)\(64\!\cdots\!52\)\( T^{2} - \)\(11\!\cdots\!72\)\( T^{3} - \)\(43\!\cdots\!30\)\( T^{4} - \)\(11\!\cdots\!72\)\( p^{13} T^{5} + \)\(64\!\cdots\!52\)\( p^{26} T^{6} + 3553486866616 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 3441261762480 T + \)\(30\!\cdots\!92\)\( T^{2} - \)\(86\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!54\)\( T^{4} - \)\(86\!\cdots\!20\)\( p^{13} T^{5} + \)\(30\!\cdots\!92\)\( p^{26} T^{6} - 3441261762480 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 2322663606000 T + \)\(78\!\cdots\!76\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!66\)\( T^{4} + \)\(12\!\cdots\!00\)\( p^{13} T^{5} + \)\(78\!\cdots\!76\)\( p^{26} T^{6} + 2322663606000 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 8462989649240 T + \)\(13\!\cdots\!08\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!74\)\( T^{4} - \)\(16\!\cdots\!40\)\( p^{13} T^{5} + \)\(13\!\cdots\!08\)\( p^{26} T^{6} - 8462989649240 p^{39} T^{7} + p^{52} T^{8} \) | |
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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
−9.063019090521418848049675744645, −8.503136455854530977399614032473, −8.487321401328237767794876212722, −8.285661335270695017557665112332, −7.73164447038168019892950631864, −7.05073564306427006151533535362, −7.00896684629682708951516429900, −6.46164967461977649957512560028, −6.26106748232255982764623335647, −6.02137491401017631859419212708, −5.36153751922669916490874015955, −5.27444069966514123280041893570, −5.17380576801744899687834141425, −4.22807913834189647493655323385, −4.11844489481388614043909528605, −4.00223498716894875950230443805, −3.53407993869415968887218564115, −2.79109301774265040591247330494, −2.67047190797960766449373613548, −1.88753900822712353920320410158, −1.79323851518260483269720167072, −1.36477817074736289249779260655, −0.951984968933101925218045970766, −0.75244812591360242009890165149, −0.69620013598661757920969419378, 0.69620013598661757920969419378, 0.75244812591360242009890165149, 0.951984968933101925218045970766, 1.36477817074736289249779260655, 1.79323851518260483269720167072, 1.88753900822712353920320410158, 2.67047190797960766449373613548, 2.79109301774265040591247330494, 3.53407993869415968887218564115, 4.00223498716894875950230443805, 4.11844489481388614043909528605, 4.22807913834189647493655323385, 5.17380576801744899687834141425, 5.27444069966514123280041893570, 5.36153751922669916490874015955, 6.02137491401017631859419212708, 6.26106748232255982764623335647, 6.46164967461977649957512560028, 7.00896684629682708951516429900, 7.05073564306427006151533535362, 7.73164447038168019892950631864, 8.285661335270695017557665112332, 8.487321401328237767794876212722, 8.503136455854530977399614032473, 9.063019090521418848049675744645