Dirichlet series
| L(s) = 1 | + 147·3-s − 1.94e3·5-s − 1.02e4·7-s − 1.74e4·9-s − 2.20e4·11-s + 1.14e5·13-s − 2.86e5·15-s − 6.96e5·17-s + 2.54e5·19-s − 1.50e6·21-s + 2.03e6·23-s − 2.00e6·25-s − 1.56e6·27-s − 6.43e6·29-s + 1.07e7·31-s − 3.23e6·33-s + 1.99e7·35-s − 3.04e7·37-s + 1.67e7·39-s − 3.39e7·41-s + 1.30e7·43-s + 3.39e7·45-s + 4.48e7·47-s − 4.46e7·49-s − 1.02e8·51-s − 1.55e8·53-s + 4.29e7·55-s + ⋯ |
| L(s) = 1 | + 1.04·3-s − 1.39·5-s − 1.61·7-s − 0.887·9-s − 0.453·11-s + 1.10·13-s − 1.45·15-s − 2.02·17-s + 0.448·19-s − 1.69·21-s + 1.51·23-s − 1.02·25-s − 0.567·27-s − 1.69·29-s + 2.08·31-s − 0.475·33-s + 2.24·35-s − 2.67·37-s + 1.16·39-s − 1.87·41-s + 0.582·43-s + 1.23·45-s + 1.34·47-s − 1.10·49-s − 2.11·51-s − 2.70·53-s + 0.632·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{16} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.31705\times 10^{8}\) |
| Root analytic conductor: | \(10.3502\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{16} \cdot 13^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | 2 | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 49 p T + 39071 T^{2} - 83248 p^{4} T^{3} + 110003644 p^{2} T^{4} - 83248 p^{13} T^{5} + 39071 p^{18} T^{6} - 49 p^{28} T^{7} + p^{36} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 1947 T + 5793193 T^{2} + 9956239494 T^{3} + 3080396895598 p T^{4} + 9956239494 p^{9} T^{5} + 5793193 p^{18} T^{6} + 1947 p^{27} T^{7} + p^{36} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + 10251 T + 21383321 p T^{2} + 21501306180 p^{2} T^{3} + 24963731669484 p^{3} T^{4} + 21501306180 p^{11} T^{5} + 21383321 p^{19} T^{6} + 10251 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 22038 T + 713066600 p T^{2} + 135630202288446 T^{3} + 26240878559314014110 T^{4} + 135630202288446 p^{9} T^{5} + 713066600 p^{19} T^{6} + 22038 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 696135 T + 308830175533 T^{2} + 69737022832404234 T^{3} + \)\(17\!\cdots\!46\)\( T^{4} + 69737022832404234 p^{9} T^{5} + 308830175533 p^{18} T^{6} + 696135 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 254502 T + 119884283960 T^{2} - 393631913266398 T^{3} + \)\(67\!\cdots\!98\)\( T^{4} - 393631913266398 p^{9} T^{5} + 119884283960 p^{18} T^{6} - 254502 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 2038992 T + 8312053042076 T^{2} - 11158305497348298768 T^{3} + \)\(10\!\cdots\!14\)\( p T^{4} - 11158305497348298768 p^{9} T^{5} + 8312053042076 p^{18} T^{6} - 2038992 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 6437112 T + 58730421223148 T^{2} + \)\(27\!\cdots\!12\)\( T^{3} + \)\(12\!\cdots\!30\)\( T^{4} + \)\(27\!\cdots\!12\)\( p^{9} T^{5} + 58730421223148 p^{18} T^{6} + 6437112 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 10739008 T + 108781000857820 T^{2} - \)\(70\!\cdots\!12\)\( T^{3} + \)\(44\!\cdots\!62\)\( T^{4} - \)\(70\!\cdots\!12\)\( p^{9} T^{5} + 108781000857820 p^{18} T^{6} - 10739008 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 30452151 T + 690929243106857 T^{2} + \)\(10\!\cdots\!78\)\( T^{3} + \)\(13\!\cdots\!62\)\( T^{4} + \)\(10\!\cdots\!78\)\( p^{9} T^{5} + 690929243106857 p^{18} T^{6} + 30452151 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 33969870 T + 1071470885942644 T^{2} + \)\(17\!\cdots\!66\)\( T^{3} + \)\(37\!\cdots\!58\)\( T^{4} + \)\(17\!\cdots\!66\)\( p^{9} T^{5} + 1071470885942644 p^{18} T^{6} + 33969870 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 13058901 T + 1197963316439687 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!16\)\( T^{4} - \)\(14\!\cdots\!80\)\( p^{9} T^{5} + 1197963316439687 p^{18} T^{6} - 13058901 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 44894985 T + 4914380269155775 T^{2} - \)\(14\!\cdots\!76\)\( T^{3} + \)\(84\!\cdots\!72\)\( T^{4} - \)\(14\!\cdots\!76\)\( p^{9} T^{5} + 4914380269155775 p^{18} T^{6} - 44894985 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 155596242 T + 11565851988449956 T^{2} + \)\(37\!\cdots\!58\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} + \)\(37\!\cdots\!58\)\( p^{9} T^{5} + 11565851988449956 p^{18} T^{6} + 155596242 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 61432458 T + 27913164001779832 T^{2} + \)\(13\!\cdots\!90\)\( T^{3} + \)\(34\!\cdots\!70\)\( T^{4} + \)\(13\!\cdots\!90\)\( p^{9} T^{5} + 27913164001779832 p^{18} T^{6} + 61432458 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 115267998 T + 27319735423367924 T^{2} - \)\(31\!\cdots\!02\)\( T^{3} + \)\(35\!\cdots\!34\)\( T^{4} - \)\(31\!\cdots\!02\)\( p^{9} T^{5} + 27319735423367924 p^{18} T^{6} - 115267998 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 406952918 T + 128923006118450104 T^{2} + \)\(31\!\cdots\!06\)\( T^{3} + \)\(54\!\cdots\!30\)\( T^{4} + \)\(31\!\cdots\!06\)\( p^{9} T^{5} + 128923006118450104 p^{18} T^{6} + 406952918 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 51911697 T + 117401414828583655 T^{2} + \)\(24\!\cdots\!12\)\( T^{3} + \)\(70\!\cdots\!56\)\( T^{4} + \)\(24\!\cdots\!12\)\( p^{9} T^{5} + 117401414828583655 p^{18} T^{6} + 51911697 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 465992648 T + 264416813138006524 T^{2} - \)\(76\!\cdots\!52\)\( T^{3} + \)\(33\!\cdots\!70\)\( p T^{4} - \)\(76\!\cdots\!52\)\( p^{9} T^{5} + 264416813138006524 p^{18} T^{6} - 465992648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 11428448 T + 106696508994002620 T^{2} - \)\(71\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!78\)\( T^{4} - \)\(71\!\cdots\!36\)\( p^{9} T^{5} + 106696508994002620 p^{18} T^{6} + 11428448 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 796564404 T + 711230362965769820 T^{2} + \)\(33\!\cdots\!92\)\( T^{3} + \)\(18\!\cdots\!02\)\( T^{4} + \)\(33\!\cdots\!92\)\( p^{9} T^{5} + 711230362965769820 p^{18} T^{6} + 796564404 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 399435336 T + 1324766398912861084 T^{2} + \)\(40\!\cdots\!08\)\( T^{3} + \)\(68\!\cdots\!54\)\( T^{4} + \)\(40\!\cdots\!08\)\( p^{9} T^{5} + 1324766398912861084 p^{18} T^{6} + 399435336 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1151600100 T + 1734183586386651332 T^{2} + \)\(13\!\cdots\!68\)\( T^{3} + \)\(15\!\cdots\!38\)\( T^{4} + \)\(13\!\cdots\!68\)\( p^{9} T^{5} + 1734183586386651332 p^{18} T^{6} + 1151600100 p^{27} T^{7} + p^{36} T^{8} \) | |
| 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
−8.089392560965300986023804761075, −7.58836672466800374029329098106, −7.49103273484320735218714981905, −7.03703270214642280307357374838, −6.90406909772694117782463823868, −6.49845681883870759332903569260, −6.42275259216685110824698955236, −6.03828010583896058817338308184, −6.00330883840261229711531100510, −5.22049937994051112848847634160, −5.18130656657903786975310514721, −4.96238418310979746334989635927, −4.56528448883236861378423035429, −3.98920619556203509468940340499, −3.88448372133089936544256269095, −3.66394077853044701307481940922, −3.48193689579208621501879896344, −2.96948841358327539120805999361, −2.85513633361335336690324180427, −2.72079087820726652180329318420, −2.42733346515425599710017991272, −1.74659079589909371093817480775, −1.45079350796811667563302180982, −1.39478606967193825361658925565, −0.68926081744090510205066289962, 0, 0, 0, 0, 0.68926081744090510205066289962, 1.39478606967193825361658925565, 1.45079350796811667563302180982, 1.74659079589909371093817480775, 2.42733346515425599710017991272, 2.72079087820726652180329318420, 2.85513633361335336690324180427, 2.96948841358327539120805999361, 3.48193689579208621501879896344, 3.66394077853044701307481940922, 3.88448372133089936544256269095, 3.98920619556203509468940340499, 4.56528448883236861378423035429, 4.96238418310979746334989635927, 5.18130656657903786975310514721, 5.22049937994051112848847634160, 6.00330883840261229711531100510, 6.03828010583896058817338308184, 6.42275259216685110824698955236, 6.49845681883870759332903569260, 6.90406909772694117782463823868, 7.03703270214642280307357374838, 7.49103273484320735218714981905, 7.58836672466800374029329098106, 8.089392560965300986023804761075