Dirichlet series
L(s) = 1 | − 3·2-s − 324·3-s − 721·4-s + 972·6-s + 9.83e3·7-s + 5.24e3·8-s + 6.56e4·9-s − 3.59e4·11-s + 2.33e5·12-s + 7.99e4·13-s − 2.95e4·14-s − 1.39e4·16-s + 6.67e5·17-s − 1.96e5·18-s − 4.25e5·19-s − 3.18e6·21-s + 1.07e5·22-s + 1.03e6·23-s − 1.69e6·24-s − 2.39e5·26-s − 1.06e7·27-s − 7.09e6·28-s + 3.67e4·29-s + 2.37e5·31-s − 2.40e6·32-s + 1.16e7·33-s − 2.00e6·34-s + ⋯ |
L(s) = 1 | − 0.132·2-s − 2.30·3-s − 1.40·4-s + 0.306·6-s + 1.54·7-s + 0.452·8-s + 10/3·9-s − 0.741·11-s + 3.25·12-s + 0.776·13-s − 0.205·14-s − 0.0530·16-s + 1.93·17-s − 0.441·18-s − 0.749·19-s − 3.57·21-s + 0.0982·22-s + 0.768·23-s − 1.04·24-s − 0.102·26-s − 3.84·27-s − 2.17·28-s + 0.00965·29-s + 0.0461·31-s − 0.405·32-s + 1.71·33-s − 0.257·34-s + ⋯ |
Functional equation
Invariants
Degree: | \(8\) |
Conductor: | \(31640625\) = \(3^{4} \cdot 5^{8}\) |
Sign: | $1$ |
Analytic conductor: | \(2.22635\times 10^{6}\) |
Root analytic conductor: | \(6.21511\) |
Motivic weight: | \(9\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((8,\ 31640625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\) |
Particular Values
\(L(5)\) | \(\approx\) | \(1.770451877\) |
\(L(\frac12)\) | \(\approx\) | \(1.770451877\) |
\(L(\frac{11}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 3 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) |
5 | \( 1 \) | ||
good | 2 | $C_2 \wr S_4$ | \( 1 + 3 T + 365 p T^{2} - 111 p^{3} T^{3} + 4077 p^{7} T^{4} - 111 p^{12} T^{5} + 365 p^{19} T^{6} + 3 p^{27} T^{7} + p^{36} T^{8} \) |
7 | $C_2 \wr S_4$ | \( 1 - 9834 T + 141893392 T^{2} - 154775191302 p T^{3} + 167640892834110 p^{2} T^{4} - 154775191302 p^{10} T^{5} + 141893392 p^{18} T^{6} - 9834 p^{27} T^{7} + p^{36} T^{8} \) | |
11 | $C_2 \wr S_4$ | \( 1 + 35994 T + 5523704672 T^{2} + 186668393765490 T^{3} + 15626722625814770286 T^{4} + 186668393765490 p^{9} T^{5} + 5523704672 p^{18} T^{6} + 35994 p^{27} T^{7} + p^{36} T^{8} \) | |
13 | $C_2 \wr S_4$ | \( 1 - 79998 T + 31802195344 T^{2} - 1495755960326586 T^{3} + \)\(42\!\cdots\!10\)\( T^{4} - 1495755960326586 p^{9} T^{5} + 31802195344 p^{18} T^{6} - 79998 p^{27} T^{7} + p^{36} T^{8} \) | |
17 | $C_2 \wr S_4$ | \( 1 - 667878 T + 282367700632 T^{2} - 126324742595722170 T^{3} + \)\(53\!\cdots\!66\)\( T^{4} - 126324742595722170 p^{9} T^{5} + 282367700632 p^{18} T^{6} - 667878 p^{27} T^{7} + p^{36} T^{8} \) | |
19 | $C_2 \wr S_4$ | \( 1 + 425792 T + 640639588492 T^{2} + 117316391220409664 T^{3} + \)\(17\!\cdots\!54\)\( T^{4} + 117316391220409664 p^{9} T^{5} + 640639588492 p^{18} T^{6} + 425792 p^{27} T^{7} + p^{36} T^{8} \) | |
23 | $C_2 \wr S_4$ | \( 1 - 1031232 T + 5668189166908 T^{2} - 5468168853855766848 T^{3} + \)\(13\!\cdots\!70\)\( T^{4} - 5468168853855766848 p^{9} T^{5} + 5668189166908 p^{18} T^{6} - 1031232 p^{27} T^{7} + p^{36} T^{8} \) | |
29 | $C_2 \wr S_4$ | \( 1 - 36786 T + 11698955610980 T^{2} + 38232367405598309058 T^{3} + \)\(21\!\cdots\!18\)\( T^{4} + 38232367405598309058 p^{9} T^{5} + 11698955610980 p^{18} T^{6} - 36786 p^{27} T^{7} + p^{36} T^{8} \) | |
31 | $C_2 \wr S_4$ | \( 1 - 237044 T + 50199397014268 T^{2} - \)\(19\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!74\)\( T^{4} - \)\(19\!\cdots\!72\)\( p^{9} T^{5} + 50199397014268 p^{18} T^{6} - 237044 p^{27} T^{7} + p^{36} T^{8} \) | |
37 | $C_2 \wr S_4$ | \( 1 - 15750594 T + 436368456033952 T^{2} - \)\(39\!\cdots\!34\)\( T^{3} + \)\(71\!\cdots\!50\)\( T^{4} - \)\(39\!\cdots\!34\)\( p^{9} T^{5} + 436368456033952 p^{18} T^{6} - 15750594 p^{27} T^{7} + p^{36} T^{8} \) | |
41 | $C_2 \wr S_4$ | \( 1 - 46660044 T + 1629598509234068 T^{2} - \)\(39\!\cdots\!32\)\( T^{3} + \)\(78\!\cdots\!54\)\( T^{4} - \)\(39\!\cdots\!32\)\( p^{9} T^{5} + 1629598509234068 p^{18} T^{6} - 46660044 p^{27} T^{7} + p^{36} T^{8} \) | |
43 | $C_2 \wr S_4$ | \( 1 - 67170720 T + 2900332889881420 T^{2} - \)\(87\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!98\)\( T^{4} - \)\(87\!\cdots\!60\)\( p^{9} T^{5} + 2900332889881420 p^{18} T^{6} - 67170720 p^{27} T^{7} + p^{36} T^{8} \) | |
47 | $C_2 \wr S_4$ | \( 1 - 48243420 T + 3581950587681820 T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(52\!\cdots\!78\)\( T^{4} - \)\(11\!\cdots\!40\)\( p^{9} T^{5} + 3581950587681820 p^{18} T^{6} - 48243420 p^{27} T^{7} + p^{36} T^{8} \) | |
53 | $C_2 \wr S_4$ | \( 1 - 198376482 T + 21847367650018168 T^{2} - \)\(16\!\cdots\!98\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} - \)\(16\!\cdots\!98\)\( p^{9} T^{5} + 21847367650018168 p^{18} T^{6} - 198376482 p^{27} T^{7} + p^{36} T^{8} \) | |
59 | $C_2 \wr S_4$ | \( 1 + 118263018 T + 24386862278408432 T^{2} + \)\(17\!\cdots\!86\)\( T^{3} + \)\(26\!\cdots\!54\)\( T^{4} + \)\(17\!\cdots\!86\)\( p^{9} T^{5} + 24386862278408432 p^{18} T^{6} + 118263018 p^{27} T^{7} + p^{36} T^{8} \) | |
61 | $C_2 \wr S_4$ | \( 1 + 178713880 T + 47272829372304076 T^{2} + \)\(59\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!06\)\( T^{4} + \)\(59\!\cdots\!20\)\( p^{9} T^{5} + 47272829372304076 p^{18} T^{6} + 178713880 p^{27} T^{7} + p^{36} T^{8} \) | |
67 | $C_2 \wr S_4$ | \( 1 - 16141548 T + 64587942934639852 T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!06\)\( T^{4} - \)\(16\!\cdots\!80\)\( p^{9} T^{5} + 64587942934639852 p^{18} T^{6} - 16141548 p^{27} T^{7} + p^{36} T^{8} \) | |
71 | $C_2 \wr S_4$ | \( 1 + 78445332 T + 132742682733898508 T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(78\!\cdots\!70\)\( T^{4} + \)\(14\!\cdots\!24\)\( p^{9} T^{5} + 132742682733898508 p^{18} T^{6} + 78445332 p^{27} T^{7} + p^{36} T^{8} \) | |
73 | $C_2 \wr S_4$ | \( 1 - 514053252 T + 188520765370005748 T^{2} - \)\(34\!\cdots\!48\)\( T^{3} + \)\(82\!\cdots\!10\)\( T^{4} - \)\(34\!\cdots\!48\)\( p^{9} T^{5} + 188520765370005748 p^{18} T^{6} - 514053252 p^{27} T^{7} + p^{36} T^{8} \) | |
79 | $C_2 \wr S_4$ | \( 1 + 431961140 T + 452072920533003676 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!66\)\( T^{4} + \)\(14\!\cdots\!80\)\( p^{9} T^{5} + 452072920533003676 p^{18} T^{6} + 431961140 p^{27} T^{7} + p^{36} T^{8} \) | |
83 | $C_2 \wr S_4$ | \( 1 - 557494176 T + 646005074335691500 T^{2} - \)\(30\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!66\)\( T^{4} - \)\(30\!\cdots\!56\)\( p^{9} T^{5} + 646005074335691500 p^{18} T^{6} - 557494176 p^{27} T^{7} + p^{36} T^{8} \) | |
89 | $C_2 \wr S_4$ | \( 1 + 178691112 T + 708605882924008892 T^{2} + \)\(39\!\cdots\!44\)\( T^{3} + \)\(23\!\cdots\!94\)\( T^{4} + \)\(39\!\cdots\!44\)\( p^{9} T^{5} + 708605882924008892 p^{18} T^{6} + 178691112 p^{27} T^{7} + p^{36} T^{8} \) | |
97 | $C_2 \wr S_4$ | \( 1 + 840904752 T + 1010444926973516932 T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(58\!\cdots\!46\)\( T^{4} + \)\(20\!\cdots\!40\)\( p^{9} T^{5} + 1010444926973516932 p^{18} T^{6} + 840904752 p^{27} T^{7} + p^{36} T^{8} \) | |
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Imaginary part of the first few zeros on the critical line
−9.200373287082590416489885943406, −8.216282383744351610890139761452, −8.172485543775915776479629521127, −8.030403381197734361316903164932, −7.66821086009572927114347316489, −7.25229706006745654740423097978, −7.01684826126683809595640566491, −6.52275362174859700813558300115, −6.20357338166374677570453656401, −5.59294248852461961521132350431, −5.56065762722815345027300034248, −5.47555390708687020794133562847, −5.14093999178245151214751952365, −4.48447872206564397506837389961, −4.19236877393513803169972526732, −4.17716236772983254094269847577, −4.17708088795468758937099736757, −3.03921690859219049061132990910, −2.72385558781020656100769999039, −2.12270992369312587219717857991, −1.63723725915114637656183449414, −1.07803679044566905166248006089, −0.934411450930870695793487568486, −0.68472310797918910595708570596, −0.32478098279394818568167629543, 0.32478098279394818568167629543, 0.68472310797918910595708570596, 0.934411450930870695793487568486, 1.07803679044566905166248006089, 1.63723725915114637656183449414, 2.12270992369312587219717857991, 2.72385558781020656100769999039, 3.03921690859219049061132990910, 4.17708088795468758937099736757, 4.17716236772983254094269847577, 4.19236877393513803169972526732, 4.48447872206564397506837389961, 5.14093999178245151214751952365, 5.47555390708687020794133562847, 5.56065762722815345027300034248, 5.59294248852461961521132350431, 6.20357338166374677570453656401, 6.52275362174859700813558300115, 7.01684826126683809595640566491, 7.25229706006745654740423097978, 7.66821086009572927114347316489, 8.030403381197734361316903164932, 8.172485543775915776479629521127, 8.216282383744351610890139761452, 9.200373287082590416489885943406