Dirichlet series
| L(s) = 1 | + 4.61e4·3-s + 7.81e6·5-s − 5.90e7·7-s − 1.24e9·9-s + 3.67e9·11-s + 7.17e9·13-s + 3.60e11·15-s − 5.67e11·17-s + 2.78e11·19-s − 2.72e12·21-s + 3.52e12·23-s + 3.81e13·25-s − 9.49e13·27-s − 6.22e13·29-s − 2.47e14·31-s + 1.69e14·33-s − 4.61e14·35-s + 7.11e14·37-s + 3.31e14·39-s − 8.68e14·41-s − 3.71e15·43-s − 9.73e15·45-s − 8.43e12·47-s − 2.90e16·49-s − 2.62e16·51-s − 4.15e16·53-s + 2.87e16·55-s + ⋯ |
| L(s) = 1 | + 1.35·3-s + 1.78·5-s − 0.553·7-s − 1.07·9-s + 0.469·11-s + 0.187·13-s + 2.42·15-s − 1.16·17-s + 0.197·19-s − 0.749·21-s + 0.408·23-s + 2·25-s − 2.39·27-s − 0.796·29-s − 1.67·31-s + 0.636·33-s − 0.989·35-s + 0.900·37-s + 0.254·39-s − 0.414·41-s − 1.12·43-s − 1.91·45-s − 0.00109·47-s − 2.54·49-s − 1.57·51-s − 1.72·53-s + 0.840·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(40960000\) = \(2^{16} \cdot 5^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.12282\times 10^{9}\) |
| Root analytic conductor: | \(13.5297\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 40960000,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\) |
Particular Values
| \(L(10)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{21}{2})\) | 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 | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 5128 p^{2} T + 125056628 p^{3} T^{2} - 18045472456 p^{8} T^{3} + 269979250617362 p^{9} T^{4} - 18045472456 p^{27} T^{5} + 125056628 p^{41} T^{6} - 5128 p^{59} T^{7} + p^{76} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 59073944 T + 4649913429070132 p T^{2} + \)\(28\!\cdots\!64\)\( p^{2} T^{3} + \)\(20\!\cdots\!34\)\( p^{4} T^{4} + \)\(28\!\cdots\!64\)\( p^{21} T^{5} + 4649913429070132 p^{39} T^{6} + 59073944 p^{57} T^{7} + p^{76} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 3675173728 T + \)\(15\!\cdots\!24\)\( T^{2} - \)\(56\!\cdots\!76\)\( p T^{3} + \)\(95\!\cdots\!50\)\( p^{2} T^{4} - \)\(56\!\cdots\!76\)\( p^{20} T^{5} + \)\(15\!\cdots\!24\)\( p^{38} T^{6} - 3675173728 p^{57} T^{7} + p^{76} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 7175523640 T + \)\(38\!\cdots\!04\)\( T^{2} - \)\(71\!\cdots\!52\)\( p T^{3} + \)\(44\!\cdots\!50\)\( p^{2} T^{4} - \)\(71\!\cdots\!52\)\( p^{20} T^{5} + \)\(38\!\cdots\!04\)\( p^{38} T^{6} - 7175523640 p^{57} T^{7} + p^{76} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 567763095896 T + \)\(22\!\cdots\!16\)\( p T^{2} + \)\(29\!\cdots\!76\)\( p^{2} T^{3} + \)\(15\!\cdots\!78\)\( p^{3} T^{4} + \)\(29\!\cdots\!76\)\( p^{21} T^{5} + \)\(22\!\cdots\!16\)\( p^{39} T^{6} + 567763095896 p^{57} T^{7} + p^{76} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 278184526832 T + \)\(19\!\cdots\!64\)\( p T^{2} - \)\(85\!\cdots\!92\)\( T^{3} + \)\(94\!\cdots\!10\)\( T^{4} - \)\(85\!\cdots\!92\)\( p^{19} T^{5} + \)\(19\!\cdots\!64\)\( p^{39} T^{6} - 278184526832 p^{57} T^{7} + p^{76} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 3524328334936 T + \)\(14\!\cdots\!20\)\( T^{2} - \)\(50\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!70\)\( T^{4} - \)\(50\!\cdots\!36\)\( p^{19} T^{5} + \)\(14\!\cdots\!20\)\( p^{38} T^{6} - 3524328334936 p^{57} T^{7} + p^{76} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 62219259151144 T + \)\(57\!\cdots\!16\)\( T^{2} - \)\(78\!\cdots\!28\)\( T^{3} + \)\(16\!\cdots\!90\)\( T^{4} - \)\(78\!\cdots\!28\)\( p^{19} T^{5} + \)\(57\!\cdots\!16\)\( p^{38} T^{6} + 62219259151144 p^{57} T^{7} + p^{76} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 247254182466832 T + \)\(33\!\cdots\!44\)\( T^{2} - \)\(17\!\cdots\!84\)\( T^{3} - \)\(62\!\cdots\!34\)\( T^{4} - \)\(17\!\cdots\!84\)\( p^{19} T^{5} + \)\(33\!\cdots\!44\)\( p^{38} T^{6} + 247254182466832 p^{57} T^{7} + p^{76} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 711533444521656 T + \)\(13\!\cdots\!92\)\( T^{2} - \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!18\)\( T^{4} - \)\(79\!\cdots\!60\)\( p^{19} T^{5} + \)\(13\!\cdots\!92\)\( p^{38} T^{6} - 711533444521656 p^{57} T^{7} + p^{76} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 868244518404600 T + \)\(12\!\cdots\!40\)\( T^{2} + \)\(15\!\cdots\!84\)\( T^{3} + \)\(66\!\cdots\!70\)\( T^{4} + \)\(15\!\cdots\!84\)\( p^{19} T^{5} + \)\(12\!\cdots\!40\)\( p^{38} T^{6} + 868244518404600 p^{57} T^{7} + p^{76} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 3716515028101432 T + \)\(42\!\cdots\!96\)\( T^{2} + \)\(10\!\cdots\!92\)\( T^{3} + \)\(67\!\cdots\!54\)\( T^{4} + \)\(10\!\cdots\!92\)\( p^{19} T^{5} + \)\(42\!\cdots\!96\)\( p^{38} T^{6} + 3716515028101432 p^{57} T^{7} + p^{76} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 8431486495352 T + \)\(15\!\cdots\!96\)\( T^{2} + \)\(60\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!94\)\( T^{4} + \)\(60\!\cdots\!36\)\( p^{19} T^{5} + \)\(15\!\cdots\!96\)\( p^{38} T^{6} + 8431486495352 p^{57} T^{7} + p^{76} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 41519794873411368 T + \)\(20\!\cdots\!52\)\( T^{2} + \)\(58\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!86\)\( T^{4} + \)\(58\!\cdots\!20\)\( p^{19} T^{5} + \)\(20\!\cdots\!52\)\( p^{38} T^{6} + 41519794873411368 p^{57} T^{7} + p^{76} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 14480346320386000 T + \)\(71\!\cdots\!32\)\( T^{2} - \)\(29\!\cdots\!64\)\( T^{3} + \)\(18\!\cdots\!06\)\( T^{4} - \)\(29\!\cdots\!64\)\( p^{19} T^{5} + \)\(71\!\cdots\!32\)\( p^{38} T^{6} + 14480346320386000 p^{57} T^{7} + p^{76} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 27840348176113048 T + \)\(13\!\cdots\!44\)\( T^{2} + \)\(83\!\cdots\!96\)\( T^{3} + \)\(59\!\cdots\!46\)\( T^{4} + \)\(83\!\cdots\!96\)\( p^{19} T^{5} + \)\(13\!\cdots\!44\)\( p^{38} T^{6} - 27840348176113048 p^{57} T^{7} + p^{76} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 8174924905481000 T + \)\(11\!\cdots\!68\)\( T^{2} - \)\(51\!\cdots\!56\)\( T^{3} + \)\(69\!\cdots\!62\)\( T^{4} - \)\(51\!\cdots\!56\)\( p^{19} T^{5} + \)\(11\!\cdots\!68\)\( p^{38} T^{6} + 8174924905481000 p^{57} T^{7} + p^{76} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 555093531720812496 T + \)\(27\!\cdots\!36\)\( T^{2} - \)\(30\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!70\)\( T^{4} - \)\(30\!\cdots\!56\)\( p^{19} T^{5} + \)\(27\!\cdots\!36\)\( p^{38} T^{6} - 555093531720812496 p^{57} T^{7} + p^{76} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 408571387318908216 T + \)\(56\!\cdots\!40\)\( T^{2} + \)\(23\!\cdots\!16\)\( T^{3} + \)\(20\!\cdots\!10\)\( T^{4} + \)\(23\!\cdots\!16\)\( p^{19} T^{5} + \)\(56\!\cdots\!40\)\( p^{38} T^{6} + 408571387318908216 p^{57} T^{7} + p^{76} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 249424427517101088 T + \)\(30\!\cdots\!44\)\( T^{2} - \)\(63\!\cdots\!92\)\( T^{3} + \)\(43\!\cdots\!70\)\( T^{4} - \)\(63\!\cdots\!92\)\( p^{19} T^{5} + \)\(30\!\cdots\!44\)\( p^{38} T^{6} - 249424427517101088 p^{57} T^{7} + p^{76} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 1696125608725987400 T + \)\(69\!\cdots\!12\)\( T^{2} - \)\(90\!\cdots\!48\)\( T^{3} + \)\(24\!\cdots\!62\)\( T^{4} - \)\(90\!\cdots\!48\)\( p^{19} T^{5} + \)\(69\!\cdots\!12\)\( p^{38} T^{6} - 1696125608725987400 p^{57} T^{7} + p^{76} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 3568552069941379992 T + \)\(37\!\cdots\!16\)\( T^{2} + \)\(88\!\cdots\!72\)\( T^{3} + \)\(56\!\cdots\!50\)\( T^{4} + \)\(88\!\cdots\!72\)\( p^{19} T^{5} + \)\(37\!\cdots\!16\)\( p^{38} T^{6} + 3568552069941379992 p^{57} T^{7} + p^{76} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 22399415502269235032 T + \)\(40\!\cdots\!16\)\( T^{2} + \)\(44\!\cdots\!16\)\( T^{3} + \)\(40\!\cdots\!74\)\( T^{4} + \)\(44\!\cdots\!16\)\( p^{19} T^{5} + \)\(40\!\cdots\!16\)\( p^{38} T^{6} + 22399415502269235032 p^{57} T^{7} + p^{76} T^{8} \) | |
| 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
−8.037320548902543650592912672855, −7.33180806408402202313790667222, −7.09316015352086248507391118359, −6.82806330726562093517830769527, −6.63989641719973587571570074125, −6.15421291864494375812637021019, −6.13611781493570646055781802893, −5.71066561785477191733579981397, −5.70181959889371728968981001173, −5.08378322573909224450548979244, −4.89426479402010630238222964566, −4.84287714724889842182861314855, −4.28119900624719379687102625172, −3.73998204859985001640725497008, −3.58951467635110978216905100102, −3.37183598717936535584800463256, −3.15081079930158268937640163008, −2.68094463686869337808001047918, −2.59837567016085317445389111376, −2.26054447924433983587128683010, −2.15190992837895273864041296222, −1.74932908495037946621832452502, −1.36384673308570639653717938900, −1.17837434055287876330843347520, −1.09553920059368461930950567558, 0, 0, 0, 0, 1.09553920059368461930950567558, 1.17837434055287876330843347520, 1.36384673308570639653717938900, 1.74932908495037946621832452502, 2.15190992837895273864041296222, 2.26054447924433983587128683010, 2.59837567016085317445389111376, 2.68094463686869337808001047918, 3.15081079930158268937640163008, 3.37183598717936535584800463256, 3.58951467635110978216905100102, 3.73998204859985001640725497008, 4.28119900624719379687102625172, 4.84287714724889842182861314855, 4.89426479402010630238222964566, 5.08378322573909224450548979244, 5.70181959889371728968981001173, 5.71066561785477191733579981397, 6.13611781493570646055781802893, 6.15421291864494375812637021019, 6.63989641719973587571570074125, 6.82806330726562093517830769527, 7.09316015352086248507391118359, 7.33180806408402202313790667222, 8.037320548902543650592912672855