Dirichlet series
| L(s) = 1 | + 3·2-s + 324·3-s − 665·4-s + 2.38e3·5-s + 972·6-s − 5.37e3·7-s − 9.23e3·8-s + 6.56e4·9-s + 7.16e3·10-s − 2.15e5·12-s + 8.83e3·13-s − 1.61e4·14-s + 7.73e5·15-s + 1.33e5·16-s − 7.06e5·17-s + 1.96e5·18-s − 9.84e5·19-s − 1.58e6·20-s − 1.74e6·21-s + 3.63e6·23-s − 2.99e6·24-s + 1.10e6·25-s + 2.65e4·26-s + 1.06e7·27-s + 3.57e6·28-s − 7.51e6·29-s + 2.32e6·30-s + ⋯ |
| L(s) = 1 | + 0.132·2-s + 2.30·3-s − 1.29·4-s + 1.70·5-s + 0.306·6-s − 0.845·7-s − 0.796·8-s + 10/3·9-s + 0.226·10-s − 2.99·12-s + 0.0858·13-s − 0.112·14-s + 3.94·15-s + 0.507·16-s − 2.05·17-s + 0.441·18-s − 1.73·19-s − 2.21·20-s − 1.95·21-s + 2.70·23-s − 1.84·24-s + 0.564·25-s + 0.0113·26-s + 3.84·27-s + 1.09·28-s − 1.97·29-s + 0.523·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(3^{4} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.22173\times 10^{9}\) |
| Root analytic conductor: | \(13.6732\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | 3 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 11 | \( 1 \) | ||
| good | 2 | $C_2 \wr S_4$ | \( 1 - 3 T + 337 p T^{2} + 163 p^{5} T^{3} + 8493 p^{5} T^{4} + 163 p^{14} T^{5} + 337 p^{19} T^{6} - 3 p^{27} T^{7} + p^{36} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 2388 T + 4600016 T^{2} - 773721964 p T^{3} + 208787129502 p^{2} T^{4} - 773721964 p^{10} T^{5} + 4600016 p^{18} T^{6} - 2388 p^{27} T^{7} + p^{36} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + 5372 T + 141869848 T^{2} + 86942078500 p T^{3} + 167654653054670 p^{2} T^{4} + 86942078500 p^{10} T^{5} + 141869848 p^{18} T^{6} + 5372 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 8836 T + 16736176816 T^{2} + 833671139248564 T^{3} + \)\(11\!\cdots\!22\)\( T^{4} + 833671139248564 p^{9} T^{5} + 16736176816 p^{18} T^{6} - 8836 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 706392 T + 32483785516 p T^{2} + 227367397847583272 T^{3} + \)\(99\!\cdots\!50\)\( T^{4} + 227367397847583272 p^{9} T^{5} + 32483785516 p^{19} T^{6} + 706392 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 984704 T + 1492801191676 T^{2} + 50153209391149440 p T^{3} + \)\(75\!\cdots\!62\)\( T^{4} + 50153209391149440 p^{10} T^{5} + 1492801191676 p^{18} T^{6} + 984704 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 3635892 T + 9119355586520 T^{2} - 17162064134202719092 T^{3} + \)\(26\!\cdots\!34\)\( T^{4} - 17162064134202719092 p^{9} T^{5} + 9119355586520 p^{18} T^{6} - 3635892 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 7517448 T + 40343949066332 T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(65\!\cdots\!78\)\( T^{4} + \)\(16\!\cdots\!00\)\( p^{9} T^{5} + 40343949066332 p^{18} T^{6} + 7517448 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 5910752 T + 99255560700220 T^{2} - \)\(41\!\cdots\!72\)\( T^{3} + \)\(38\!\cdots\!82\)\( T^{4} - \)\(41\!\cdots\!72\)\( p^{9} T^{5} + 99255560700220 p^{18} T^{6} - 5910752 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 7163752 T + 264533501074156 T^{2} + \)\(16\!\cdots\!84\)\( T^{3} + \)\(38\!\cdots\!62\)\( T^{4} + \)\(16\!\cdots\!84\)\( p^{9} T^{5} + 264533501074156 p^{18} T^{6} + 7163752 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 596832 T + 717504170551820 T^{2} + \)\(18\!\cdots\!68\)\( T^{3} + \)\(25\!\cdots\!82\)\( T^{4} + \)\(18\!\cdots\!68\)\( p^{9} T^{5} + 717504170551820 p^{18} T^{6} - 596832 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 14430328 T + 1031248427309164 T^{2} + \)\(24\!\cdots\!72\)\( T^{3} + \)\(39\!\cdots\!90\)\( T^{4} + \)\(24\!\cdots\!72\)\( p^{9} T^{5} + 1031248427309164 p^{18} T^{6} - 14430328 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 6723588 T + 1556533169584184 T^{2} + \)\(21\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!46\)\( T^{4} + \)\(21\!\cdots\!24\)\( p^{9} T^{5} + 1556533169584184 p^{18} T^{6} + 6723588 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 69704508 T + 7126861173287024 T^{2} + \)\(27\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!62\)\( T^{4} + \)\(27\!\cdots\!04\)\( p^{9} T^{5} + 7126861173287024 p^{18} T^{6} + 69704508 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 13672392 T + 17094885069049244 T^{2} - \)\(59\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!42\)\( T^{4} - \)\(59\!\cdots\!88\)\( p^{9} T^{5} + 17094885069049244 p^{18} T^{6} - 13672392 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 88290604 T + 17888377972117504 T^{2} + \)\(12\!\cdots\!48\)\( T^{3} - \)\(22\!\cdots\!42\)\( p T^{4} + \)\(12\!\cdots\!48\)\( p^{9} T^{5} + 17888377972117504 p^{18} T^{6} - 88290604 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 402240496 T + 128229260106826924 T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!14\)\( T^{4} + \)\(26\!\cdots\!20\)\( p^{9} T^{5} + 128229260106826924 p^{18} T^{6} + 402240496 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 474595140 T + 147939394104968024 T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + \)\(52\!\cdots\!22\)\( T^{4} - \)\(26\!\cdots\!60\)\( p^{9} T^{5} + 147939394104968024 p^{18} T^{6} - 474595140 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 425419592 T + 152815803450241468 T^{2} + \)\(34\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(34\!\cdots\!08\)\( p^{9} T^{5} + 152815803450241468 p^{18} T^{6} + 425419592 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 538408868 T + 512485077024820312 T^{2} + \)\(18\!\cdots\!60\)\( T^{3} + \)\(92\!\cdots\!18\)\( T^{4} + \)\(18\!\cdots\!60\)\( p^{9} T^{5} + 512485077024820312 p^{18} T^{6} + 538408868 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 783087552 T + 968601818452788140 T^{2} + \)\(46\!\cdots\!20\)\( T^{3} + \)\(29\!\cdots\!70\)\( T^{4} + \)\(46\!\cdots\!20\)\( p^{9} T^{5} + 968601818452788140 p^{18} T^{6} + 783087552 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 747115224 T + 1272286980074069084 T^{2} - \)\(71\!\cdots\!92\)\( T^{3} + \)\(64\!\cdots\!86\)\( T^{4} - \)\(71\!\cdots\!92\)\( p^{9} T^{5} + 1272286980074069084 p^{18} T^{6} - 747115224 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 814209296 T + 2528995424885977564 T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!98\)\( T^{4} - \)\(13\!\cdots\!56\)\( p^{9} T^{5} + 2528995424885977564 p^{18} T^{6} - 814209296 p^{27} T^{7} + p^{36} T^{8} \) | |
| 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
−7.12510536151920846430640439450, −6.85232825431308770717757573384, −6.70510226352948636466696398973, −6.68168781359794039878196232938, −6.13791061527055014337088892615, −6.10791725184272549339569534639, −5.86552483929369227996232359274, −5.31482290327833556785067140733, −5.18073355556569740413199596558, −4.86089618343665177606625095945, −4.47403079451446319357606540824, −4.42370173176954407118766464362, −4.30416098834105497914653336026, −3.66375603876532230964753794373, −3.66229204540631020330319659135, −3.33259229407357798741625161239, −2.93958900975060268226133479522, −2.80768710500965960473512567053, −2.59255843062428544094255048056, −2.21441302188474677553528557186, −2.07684843526289066853536350534, −1.81829898495181398467993224806, −1.36473917838542009265992253976, −1.13368116814014446978527401461, −1.10677134513827314714893625302, 0, 0, 0, 0, 1.10677134513827314714893625302, 1.13368116814014446978527401461, 1.36473917838542009265992253976, 1.81829898495181398467993224806, 2.07684843526289066853536350534, 2.21441302188474677553528557186, 2.59255843062428544094255048056, 2.80768710500965960473512567053, 2.93958900975060268226133479522, 3.33259229407357798741625161239, 3.66229204540631020330319659135, 3.66375603876532230964753794373, 4.30416098834105497914653336026, 4.42370173176954407118766464362, 4.47403079451446319357606540824, 4.86089618343665177606625095945, 5.18073355556569740413199596558, 5.31482290327833556785067140733, 5.86552483929369227996232359274, 6.10791725184272549339569534639, 6.13791061527055014337088892615, 6.68168781359794039878196232938, 6.70510226352948636466696398973, 6.85232825431308770717757573384, 7.12510536151920846430640439450