Dirichlet series
| L(s) = 1 | + 324·3-s + 1.72e3·5-s + 4.84e3·7-s + 6.56e4·9-s + 1.58e4·11-s + 8.24e4·13-s + 5.59e5·15-s + 1.65e5·17-s + 5.39e5·19-s + 1.56e6·21-s + 7.29e5·23-s − 2.30e6·25-s + 1.06e7·27-s + 1.85e6·29-s + 3.19e6·31-s + 5.12e6·33-s + 8.36e6·35-s + 4.18e6·37-s + 2.67e7·39-s + 2.27e5·41-s + 1.60e6·43-s + 1.13e8·45-s + 1.80e7·47-s − 4.01e7·49-s + 5.37e7·51-s + 2.94e7·53-s + 2.73e7·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.23·5-s + 0.761·7-s + 10/3·9-s + 0.325·11-s + 0.800·13-s + 2.85·15-s + 0.481·17-s + 0.950·19-s + 1.75·21-s + 0.543·23-s − 1.17·25-s + 3.84·27-s + 0.485·29-s + 0.621·31-s + 0.752·33-s + 0.942·35-s + 0.367·37-s + 1.84·39-s + 0.0125·41-s + 0.0713·43-s + 4.12·45-s + 0.539·47-s − 0.994·49-s + 1.11·51-s + 0.512·53-s + 0.402·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{28} \cdot 3^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.52994\times 10^{9}\) |
| Root analytic conductor: | \(14.0632\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(72.39871931\) |
| \(L(\frac12)\) | \(\approx\) | \(72.39871931\) |
| \(L(\frac{11}{2})\) | 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 | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 1728 T + 5286836 T^{2} - 1344360512 p T^{3} + 585667235526 p^{2} T^{4} - 1344360512 p^{10} T^{5} + 5286836 p^{18} T^{6} - 1728 p^{27} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 4840 T + 9079372 p T^{2} - 9122761384 p^{2} T^{3} + 9758964239914 p^{3} T^{4} - 9122761384 p^{11} T^{5} + 9079372 p^{19} T^{6} - 4840 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 15824 T + 3272030348 T^{2} - 110832700706512 T^{3} + 11168899925921994134 T^{4} - 110832700706512 p^{9} T^{5} + 3272030348 p^{18} T^{6} - 15824 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 82440 T + 32507843788 T^{2} - 2806427918898776 T^{3} + \)\(46\!\cdots\!74\)\( T^{4} - 2806427918898776 p^{9} T^{5} + 32507843788 p^{18} T^{6} - 82440 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 165912 T + 402742640540 T^{2} - 57573640941296168 T^{3} + \)\(68\!\cdots\!90\)\( T^{4} - 57573640941296168 p^{9} T^{5} + 402742640540 p^{18} T^{6} - 165912 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 28416 p T + 878148739660 T^{2} - 411395194240546048 T^{3} + \)\(35\!\cdots\!02\)\( T^{4} - 411395194240546048 p^{9} T^{5} + 878148739660 p^{18} T^{6} - 28416 p^{28} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 729680 T + 3870147116348 T^{2} - 3090843717350364304 T^{3} + \)\(76\!\cdots\!54\)\( T^{4} - 3090843717350364304 p^{9} T^{5} + 3870147116348 p^{18} T^{6} - 729680 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 1850864 T + 51896421352052 T^{2} - 76884864756743058576 T^{3} + \)\(10\!\cdots\!42\)\( T^{4} - 76884864756743058576 p^{9} T^{5} + 51896421352052 p^{18} T^{6} - 1850864 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 103160 p T + 80766069413044 T^{2} - \)\(19\!\cdots\!92\)\( T^{3} + \)\(28\!\cdots\!74\)\( T^{4} - \)\(19\!\cdots\!92\)\( p^{9} T^{5} + 80766069413044 p^{18} T^{6} - 103160 p^{28} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 4187992 T + 316603511258092 T^{2} - \)\(22\!\cdots\!72\)\( T^{3} + \)\(48\!\cdots\!50\)\( T^{4} - \)\(22\!\cdots\!72\)\( p^{9} T^{5} + 316603511258092 p^{18} T^{6} - 4187992 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 227704 T + 815482715263292 T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!46\)\( T^{4} - \)\(24\!\cdots\!80\)\( p^{9} T^{5} + 815482715263292 p^{18} T^{6} - 227704 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 1600352 T + 1039425217732396 T^{2} + \)\(25\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!14\)\( T^{4} + \)\(25\!\cdots\!40\)\( p^{9} T^{5} + 1039425217732396 p^{18} T^{6} - 1600352 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 18053904 T + 898850271089564 T^{2} - \)\(31\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!54\)\( T^{4} - \)\(31\!\cdots\!80\)\( p^{9} T^{5} + 898850271089564 p^{18} T^{6} - 18053904 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 29418288 T + 8246475311053268 T^{2} - \)\(75\!\cdots\!52\)\( T^{3} + \)\(31\!\cdots\!10\)\( T^{4} - \)\(75\!\cdots\!52\)\( p^{9} T^{5} + 8246475311053268 p^{18} T^{6} - 29418288 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 38300048 T + 12480247782213068 T^{2} - \)\(64\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!66\)\( T^{4} - \)\(64\!\cdots\!60\)\( p^{9} T^{5} + 12480247782213068 p^{18} T^{6} - 38300048 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 99764648 T + 35070113945959756 T^{2} + \)\(25\!\cdots\!08\)\( T^{3} + \)\(53\!\cdots\!54\)\( T^{4} + \)\(25\!\cdots\!08\)\( p^{9} T^{5} + 35070113945959756 p^{18} T^{6} + 99764648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 183717008 T + 94982362218124012 T^{2} + \)\(13\!\cdots\!36\)\( T^{3} + \)\(56\!\cdots\!98\)\( p T^{4} + \)\(13\!\cdots\!36\)\( p^{9} T^{5} + 94982362218124012 p^{18} T^{6} + 183717008 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 181868080 T + 187559006549429756 T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!80\)\( p^{9} T^{5} + 187559006549429756 p^{18} T^{6} - 181868080 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 254539160 T + 208692556022844412 T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!82\)\( T^{4} - \)\(39\!\cdots\!64\)\( p^{9} T^{5} + 208692556022844412 p^{18} T^{6} - 254539160 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 831578184 T + 541079473633225972 T^{2} - \)\(23\!\cdots\!64\)\( T^{3} + \)\(93\!\cdots\!90\)\( T^{4} - \)\(23\!\cdots\!64\)\( p^{9} T^{5} + 541079473633225972 p^{18} T^{6} - 831578184 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 687923952 T + 569668186782399404 T^{2} + \)\(18\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!50\)\( T^{4} + \)\(18\!\cdots\!04\)\( p^{9} T^{5} + 569668186782399404 p^{18} T^{6} + 687923952 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 627627272 T + 1209510335623979132 T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(60\!\cdots\!14\)\( T^{4} - \)\(64\!\cdots\!44\)\( p^{9} T^{5} + 1209510335623979132 p^{18} T^{6} - 627627272 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 889385880 T + 757736193785874268 T^{2} + \)\(68\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} + \)\(68\!\cdots\!80\)\( p^{9} T^{5} + 757736193785874268 p^{18} T^{6} + 889385880 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−6.61850662794234907328942965591, −6.37215697482368782477628354805, −6.19002330582254247129061477360, −6.13331777737793219390281676115, −5.67890540998574231513375271654, −5.26486742461106083882064829493, −5.13073777859526291827381865642, −5.00064938426459951183435895809, −4.62530701773421786742790593790, −4.14783636144836508997256069437, −4.05322848445806935175790153026, −3.75492545150283330551566731300, −3.71215699096809311808831218894, −3.01435676038635351805363577479, −2.98467667422709821432920041060, −2.92760642540838768347961692117, −2.55055231197873506217579348191, −1.97987009520064533181380419620, −1.91689447585586066955072201677, −1.82084635139231384387782673966, −1.58285604268101640001852821973, −1.08407537519477979429015482689, −0.975492411677741652025186250866, −0.63785976225632888097147145255, −0.40407642548045231245487673679, 0.40407642548045231245487673679, 0.63785976225632888097147145255, 0.975492411677741652025186250866, 1.08407537519477979429015482689, 1.58285604268101640001852821973, 1.82084635139231384387782673966, 1.91689447585586066955072201677, 1.97987009520064533181380419620, 2.55055231197873506217579348191, 2.92760642540838768347961692117, 2.98467667422709821432920041060, 3.01435676038635351805363577479, 3.71215699096809311808831218894, 3.75492545150283330551566731300, 4.05322848445806935175790153026, 4.14783636144836508997256069437, 4.62530701773421786742790593790, 5.00064938426459951183435895809, 5.13073777859526291827381865642, 5.26486742461106083882064829493, 5.67890540998574231513375271654, 6.13331777737793219390281676115, 6.19002330582254247129061477360, 6.37215697482368782477628354805, 6.61850662794234907328942965591