Dirichlet series
| L(s) = 1 | − 64·2-s − 324·3-s + 2.56e3·4-s + 420·5-s + 2.07e4·6-s + 1.85e3·7-s − 8.19e4·8-s + 6.56e4·9-s − 2.68e4·10-s + 9.55e4·11-s − 8.29e5·12-s + 7.02e4·13-s − 1.18e5·14-s − 1.36e5·15-s + 2.29e6·16-s + 3.69e5·17-s − 4.19e6·18-s + 5.21e5·19-s + 1.07e6·20-s − 6.00e5·21-s − 6.11e6·22-s − 5.91e5·23-s + 2.65e7·24-s − 3.92e6·25-s − 4.49e6·26-s − 1.06e7·27-s + 4.74e6·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 0.300·5-s + 6.53·6-s + 0.291·7-s − 7.07·8-s + 10/3·9-s − 0.850·10-s + 1.96·11-s − 11.5·12-s + 0.682·13-s − 0.825·14-s − 0.694·15-s + 35/4·16-s + 1.07·17-s − 9.42·18-s + 0.917·19-s + 1.50·20-s − 0.674·21-s − 5.56·22-s − 0.440·23-s + 16.3·24-s − 2.01·25-s − 1.93·26-s − 3.84·27-s + 1.45·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 19^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18841\times 10^{7}\) |
| Root analytic conductor: | \(7.66251\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(0.8720460455\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8720460455\) |
| \(L(\frac{11}{2})\) | 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 | 2 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 84 p T + 4102529 T^{2} - 361586718 p T^{3} + 2134042766616 p T^{4} - 361586718 p^{10} T^{5} + 4102529 p^{18} T^{6} - 84 p^{28} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 1854 T + 4976963 p T^{2} + 2140816302 p^{2} T^{3} - 1088243791620 p^{3} T^{4} + 2140816302 p^{11} T^{5} + 4976963 p^{19} T^{6} - 1854 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 95570 T + 6596883693 T^{2} - 202181470808238 T^{3} + 9480823760686186636 T^{4} - 202181470808238 p^{9} T^{5} + 6596883693 p^{18} T^{6} - 95570 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 70294 T + 17224415072 T^{2} - 990342171637986 T^{3} + \)\(24\!\cdots\!54\)\( T^{4} - 990342171637986 p^{9} T^{5} + 17224415072 p^{18} T^{6} - 70294 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 369878 T + 249971462361 T^{2} - 7523008918407114 p T^{3} + \)\(34\!\cdots\!04\)\( T^{4} - 7523008918407114 p^{10} T^{5} + 249971462361 p^{18} T^{6} - 369878 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 591538 T + 4600469022060 T^{2} + 3026126608599892890 T^{3} + \)\(21\!\cdots\!86\)\( p^{2} T^{4} + 3026126608599892890 p^{9} T^{5} + 4600469022060 p^{18} T^{6} + 591538 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 5453976 T + 42256646485340 T^{2} + \)\(19\!\cdots\!56\)\( T^{3} + \)\(87\!\cdots\!58\)\( T^{4} + \)\(19\!\cdots\!56\)\( p^{9} T^{5} + 42256646485340 p^{18} T^{6} + 5453976 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 7099154 T + 94396920486848 T^{2} + \)\(46\!\cdots\!78\)\( T^{3} + \)\(35\!\cdots\!14\)\( T^{4} + \)\(46\!\cdots\!78\)\( p^{9} T^{5} + 94396920486848 p^{18} T^{6} + 7099154 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 28820022 T + 693896775088388 T^{2} + \)\(10\!\cdots\!26\)\( T^{3} + \)\(14\!\cdots\!94\)\( T^{4} + \)\(10\!\cdots\!26\)\( p^{9} T^{5} + 693896775088388 p^{18} T^{6} + 28820022 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 44289304 T + 1938393631722672 T^{2} + \)\(46\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!42\)\( T^{4} + \)\(46\!\cdots\!92\)\( p^{9} T^{5} + 1938393631722672 p^{18} T^{6} + 44289304 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 10650682 T + 1593340682210045 T^{2} - \)\(13\!\cdots\!50\)\( T^{3} + \)\(10\!\cdots\!64\)\( T^{4} - \)\(13\!\cdots\!50\)\( p^{9} T^{5} + 1593340682210045 p^{18} T^{6} - 10650682 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 32468044 T + 1932285324719601 T^{2} + \)\(47\!\cdots\!94\)\( T^{3} + \)\(19\!\cdots\!56\)\( T^{4} + \)\(47\!\cdots\!94\)\( p^{9} T^{5} + 1932285324719601 p^{18} T^{6} + 32468044 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 11463536 T + 7193363743814556 T^{2} - \)\(11\!\cdots\!36\)\( T^{3} + \)\(24\!\cdots\!18\)\( T^{4} - \)\(11\!\cdots\!36\)\( p^{9} T^{5} + 7193363743814556 p^{18} T^{6} + 11463536 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 56820680 T + 33147984722744508 T^{2} - \)\(14\!\cdots\!32\)\( T^{3} + \)\(42\!\cdots\!02\)\( T^{4} - \)\(14\!\cdots\!32\)\( p^{9} T^{5} + 33147984722744508 p^{18} T^{6} - 56820680 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 161703434 T + 51578219363281273 T^{2} - \)\(56\!\cdots\!22\)\( T^{3} + \)\(93\!\cdots\!24\)\( T^{4} - \)\(56\!\cdots\!22\)\( p^{9} T^{5} + 51578219363281273 p^{18} T^{6} - 161703434 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 190008892 T + 89492694671583308 T^{2} - \)\(12\!\cdots\!72\)\( T^{3} + \)\(34\!\cdots\!34\)\( T^{4} - \)\(12\!\cdots\!72\)\( p^{9} T^{5} + 89492694671583308 p^{18} T^{6} - 190008892 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 475310824 T + 207364254911992812 T^{2} - \)\(62\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!06\)\( T^{4} - \)\(62\!\cdots\!60\)\( p^{9} T^{5} + 207364254911992812 p^{18} T^{6} - 475310824 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 578125122 T + 211593840720388121 T^{2} - \)\(45\!\cdots\!86\)\( T^{3} + \)\(10\!\cdots\!84\)\( T^{4} - \)\(45\!\cdots\!86\)\( p^{9} T^{5} + 211593840720388121 p^{18} T^{6} - 578125122 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 1141283142 T + 827560589725394996 T^{2} - \)\(42\!\cdots\!94\)\( T^{3} + \)\(16\!\cdots\!26\)\( T^{4} - \)\(42\!\cdots\!94\)\( p^{9} T^{5} + 827560589725394996 p^{18} T^{6} - 1141283142 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 841631224 T + 792443870052979164 T^{2} - \)\(40\!\cdots\!36\)\( T^{3} + \)\(21\!\cdots\!94\)\( T^{4} - \)\(40\!\cdots\!36\)\( p^{9} T^{5} + 792443870052979164 p^{18} T^{6} - 841631224 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 112454896 T + 992224031354080380 T^{2} + \)\(83\!\cdots\!48\)\( T^{3} + \)\(48\!\cdots\!66\)\( T^{4} + \)\(83\!\cdots\!48\)\( p^{9} T^{5} + 992224031354080380 p^{18} T^{6} + 112454896 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 2201574500 T + 3440032038621633844 T^{2} - \)\(40\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!18\)\( T^{4} - \)\(40\!\cdots\!80\)\( p^{9} T^{5} + 3440032038621633844 p^{18} T^{6} - 2201574500 p^{27} T^{7} + p^{36} T^{8} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.310666311717367539385328327606, −7.73963916923366708418721622312, −7.62067607934007527069881520899, −7.41136554258741441313323541232, −7.09923613238021705093497336040, −6.53574932160812664772383493715, −6.50651743531986979314919654735, −6.39224126145135007064561465694, −6.13782151835130093601684166358, −5.50755144686916248943720792901, −5.34291766334547658931564484108, −5.17215910188162511409561030550, −4.92131802520139692402398704540, −3.86427252114670214682452629637, −3.62422210220377052294439003633, −3.56503086514829096390425385079, −3.46067881326640140869790293805, −2.07545567326309319126209762139, −2.00453589631803190106495725419, −1.77823012794882042077875200155, −1.63912216800227749377137116352, −1.09507760453938210157665279895, −0.71751754822906277537718180854, −0.46897863258085875995213191659, −0.42405960748720443954102979264, 0.42405960748720443954102979264, 0.46897863258085875995213191659, 0.71751754822906277537718180854, 1.09507760453938210157665279895, 1.63912216800227749377137116352, 1.77823012794882042077875200155, 2.00453589631803190106495725419, 2.07545567326309319126209762139, 3.46067881326640140869790293805, 3.56503086514829096390425385079, 3.62422210220377052294439003633, 3.86427252114670214682452629637, 4.92131802520139692402398704540, 5.17215910188162511409561030550, 5.34291766334547658931564484108, 5.50755144686916248943720792901, 6.13782151835130093601684166358, 6.39224126145135007064561465694, 6.50651743531986979314919654735, 6.53574932160812664772383493715, 7.09923613238021705093497336040, 7.41136554258741441313323541232, 7.62067607934007527069881520899, 7.73963916923366708418721622312, 8.310666311717367539385328327606