Dirichlet series
| L(s) = 1 | − 324·3-s + 240·5-s + 4.84e3·7-s + 6.56e4·9-s + 9.96e4·11-s + 6.08e4·13-s − 7.77e4·15-s − 4.34e5·17-s − 6.31e5·19-s − 1.56e6·21-s + 7.49e5·23-s − 8.81e5·25-s − 1.06e7·27-s + 7.90e6·29-s + 1.13e7·31-s − 3.22e7·33-s + 1.16e6·35-s + 1.35e7·37-s − 1.97e7·39-s − 1.88e7·41-s − 1.41e7·43-s + 1.57e7·45-s − 3.77e7·47-s − 6.42e7·49-s + 1.40e8·51-s − 1.15e8·53-s + 2.39e7·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.171·5-s + 0.761·7-s + 10/3·9-s + 2.05·11-s + 0.590·13-s − 0.396·15-s − 1.26·17-s − 1.11·19-s − 1.75·21-s + 0.558·23-s − 0.451·25-s − 3.84·27-s + 2.07·29-s + 2.20·31-s − 4.73·33-s + 0.130·35-s + 1.19·37-s − 1.36·39-s − 1.04·41-s − 0.632·43-s + 0.572·45-s − 1.12·47-s − 1.59·49-s + 2.91·51-s − 2.00·53-s + 0.352·55-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
| \(L(5)\) | \(\approx\) | \(2.354616452\) |
| \(L(\frac12)\) | \(\approx\) | \(2.354616452\) |
| \(L(\frac{11}{2})\) | 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 | $C_2 \wr S_4$ | \( 1 - 48 p T + 939284 T^{2} + 236450096 p T^{3} - 167112133242 p^{2} T^{4} + 236450096 p^{10} T^{5} + 939284 p^{18} T^{6} - 48 p^{28} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 4840 T + 12530764 p T^{2} - 12761367208 p^{2} T^{3} + 11457097294378 p^{3} T^{4} - 12761367208 p^{11} T^{5} + 12530764 p^{19} T^{6} - 4840 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 99664 T + 10589400716 T^{2} - 699518969642576 T^{3} + 38813237732845803926 T^{4} - 699518969642576 p^{9} T^{5} + 10589400716 p^{18} T^{6} - 99664 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 360 p^{2} T + 24714128908 T^{2} - 695713226418488 T^{3} + \)\(29\!\cdots\!86\)\( T^{4} - 695713226418488 p^{9} T^{5} + 24714128908 p^{18} T^{6} - 360 p^{29} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 434952 T + 394190258396 T^{2} + 122601307335261368 T^{3} + \)\(64\!\cdots\!26\)\( T^{4} + 122601307335261368 p^{9} T^{5} + 394190258396 p^{18} T^{6} + 434952 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 631776 T + 1164722951116 T^{2} + 471204029388957920 T^{3} + \)\(52\!\cdots\!86\)\( T^{4} + 471204029388957920 p^{9} T^{5} + 1164722951116 p^{18} T^{6} + 631776 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 749392 T + 3347296061756 T^{2} - 4837140337367041424 T^{3} + \)\(64\!\cdots\!18\)\( T^{4} - 4837140337367041424 p^{9} T^{5} + 3347296061756 p^{18} T^{6} - 749392 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 7908544 T + 58432091245844 T^{2} - \)\(22\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} - \)\(22\!\cdots\!08\)\( p^{9} T^{5} + 58432091245844 p^{18} T^{6} - 7908544 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 11351240 T + 115399090325812 T^{2} - \)\(71\!\cdots\!24\)\( T^{3} + \)\(43\!\cdots\!22\)\( T^{4} - \)\(71\!\cdots\!24\)\( p^{9} T^{5} + 115399090325812 p^{18} T^{6} - 11351240 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 13592920 T + 213368728836076 T^{2} + 87591136825729972088 T^{3} + \)\(27\!\cdots\!94\)\( T^{4} + 87591136825729972088 p^{9} T^{5} + 213368728836076 p^{18} T^{6} - 13592920 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 18838888 T + 656876228382716 T^{2} + \)\(15\!\cdots\!88\)\( T^{3} + \)\(73\!\cdots\!90\)\( p T^{4} + \)\(15\!\cdots\!88\)\( p^{9} T^{5} + 656876228382716 p^{18} T^{6} + 18838888 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 14177920 T + 1417437436848556 T^{2} + \)\(23\!\cdots\!96\)\( T^{3} + \)\(93\!\cdots\!54\)\( T^{4} + \)\(23\!\cdots\!96\)\( p^{9} T^{5} + 1417437436848556 p^{18} T^{6} + 14177920 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 37779120 T + 3062567184137372 T^{2} + \)\(92\!\cdots\!16\)\( T^{3} + \)\(49\!\cdots\!30\)\( T^{4} + \)\(92\!\cdots\!16\)\( p^{9} T^{5} + 3062567184137372 p^{18} T^{6} + 37779120 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 115336512 T + 10787477606282612 T^{2} + \)\(79\!\cdots\!64\)\( T^{3} + \)\(46\!\cdots\!26\)\( T^{4} + \)\(79\!\cdots\!64\)\( p^{9} T^{5} + 10787477606282612 p^{18} T^{6} + 115336512 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 115028080 T + 604137837099172 p T^{2} - \)\(28\!\cdots\!56\)\( T^{3} + \)\(46\!\cdots\!30\)\( T^{4} - \)\(28\!\cdots\!56\)\( p^{9} T^{5} + 604137837099172 p^{19} T^{6} - 115028080 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 173228648 T + 46925341200786892 T^{2} + \)\(58\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!42\)\( T^{4} + \)\(58\!\cdots\!20\)\( p^{9} T^{5} + 46925341200786892 p^{18} T^{6} + 173228648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 231785104 T + 47923090341550828 T^{2} - \)\(16\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!82\)\( T^{4} - \)\(16\!\cdots\!56\)\( p^{9} T^{5} + 47923090341550828 p^{18} T^{6} - 231785104 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 197476208 T + 67580246245585148 T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!06\)\( T^{4} - \)\(16\!\cdots\!80\)\( p^{9} T^{5} + 67580246245585148 p^{18} T^{6} - 197476208 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 44629400 T + 155814395021474812 T^{2} - \)\(40\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!82\)\( T^{4} - \)\(40\!\cdots\!56\)\( p^{9} T^{5} + 155814395021474812 p^{18} T^{6} - 44629400 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 355774584 T + 431793154567003252 T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!98\)\( T^{4} + \)\(12\!\cdots\!20\)\( p^{9} T^{5} + 431793154567003252 p^{18} T^{6} + 355774584 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 607613328 T + 689233408599137708 T^{2} - \)\(31\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!34\)\( T^{4} - \)\(31\!\cdots\!68\)\( p^{9} T^{5} + 689233408599137708 p^{18} T^{6} - 607613328 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 1157146424 T + 1248537877761010940 T^{2} + \)\(74\!\cdots\!24\)\( T^{3} + \)\(52\!\cdots\!78\)\( T^{4} + \)\(74\!\cdots\!24\)\( p^{9} T^{5} + 1248537877761010940 p^{18} T^{6} + 1157146424 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1599536472 T + 3875436244314154204 T^{2} + \)\(37\!\cdots\!12\)\( T^{3} + \)\(47\!\cdots\!58\)\( T^{4} + \)\(37\!\cdots\!12\)\( p^{9} T^{5} + 3875436244314154204 p^{18} T^{6} + 1599536472 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−6.69305774727702006001489278630, −6.39287817832682821440961234313, −6.19947333254074990508588305548, −6.07136003387312722089378991671, −5.89716023797682398583581943233, −5.28229962341341675472966776488, −5.05097220081053951928841175064, −4.99244312008767037811188636567, −4.71995788765678205806170904210, −4.50293124738445231482510804846, −4.12525058510336172980411296671, −4.05052466920498709014475812946, −3.98359772705021390675097038581, −3.31049197798413088209502059114, −2.99579853574555327228108902678, −2.84264302932600130754439240219, −2.43639688377719806146342068964, −1.80141486670372477889577922678, −1.75553476823365096176744501935, −1.55238467777776968254714133599, −1.36510112690037064318345365199, −0.927787026493126033602394674716, −0.64913267999754629003695635287, −0.58201593581893872336733478882, −0.18704348531330989235488134317, 0.18704348531330989235488134317, 0.58201593581893872336733478882, 0.64913267999754629003695635287, 0.927787026493126033602394674716, 1.36510112690037064318345365199, 1.55238467777776968254714133599, 1.75553476823365096176744501935, 1.80141486670372477889577922678, 2.43639688377719806146342068964, 2.84264302932600130754439240219, 2.99579853574555327228108902678, 3.31049197798413088209502059114, 3.98359772705021390675097038581, 4.05052466920498709014475812946, 4.12525058510336172980411296671, 4.50293124738445231482510804846, 4.71995788765678205806170904210, 4.99244312008767037811188636567, 5.05097220081053951928841175064, 5.28229962341341675472966776488, 5.89716023797682398583581943233, 6.07136003387312722089378991671, 6.19947333254074990508588305548, 6.39287817832682821440961234313, 6.69305774727702006001489278630