Dirichlet series
| L(s) = 1 | + 46·2-s + 972·3-s − 714·4-s + 4.47e4·6-s + 6.83e4·7-s + 3.20e4·8-s + 5.90e5·9-s − 9.99e4·11-s − 6.94e5·12-s + 2.30e6·13-s + 3.14e6·14-s + 3.88e6·16-s − 3.44e6·17-s + 2.71e7·18-s − 4.21e6·19-s + 6.64e7·21-s − 4.59e6·22-s + 5.26e7·23-s + 3.11e7·24-s + 1.06e8·26-s + 2.86e8·27-s − 4.88e7·28-s + 2.17e8·29-s − 3.26e8·31-s − 4.65e7·32-s − 9.71e7·33-s − 1.58e8·34-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 2.30·3-s − 0.348·4-s + 2.34·6-s + 1.53·7-s + 0.345·8-s + 10/3·9-s − 0.187·11-s − 0.805·12-s + 1.72·13-s + 1.56·14-s + 0.926·16-s − 0.588·17-s + 3.38·18-s − 0.390·19-s + 3.55·21-s − 0.190·22-s + 1.70·23-s + 0.798·24-s + 1.75·26-s + 3.84·27-s − 0.536·28-s + 1.96·29-s − 2.04·31-s − 0.245·32-s − 0.432·33-s − 0.597·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(31640625\) = \(3^{4} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.10272\times 10^{7}\) |
| Root analytic conductor: | \(7.59116\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 31640625,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(52.08821569\) |
| \(L(\frac12)\) | \(\approx\) | \(52.08821569\) |
| \(L(\frac{13}{2})\) | 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 | 3 | $C_1$ | \( ( 1 - p^{5} T )^{4} \) |
| 5 | \( 1 \) | ||
| good | 2 | $C_2 \wr S_4$ | \( 1 - 23 p T + 1415 p T^{2} - 12191 p^{4} T^{3} + 134099 p^{6} T^{4} - 12191 p^{15} T^{5} + 1415 p^{23} T^{6} - 23 p^{34} T^{7} + p^{44} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 68372 T + 3685360978 T^{2} - 22270114394664 p T^{3} + 185231300643789915 p^{2} T^{4} - 22270114394664 p^{12} T^{5} + 3685360978 p^{22} T^{6} - 68372 p^{33} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 99944 T + 3829207972 p^{2} T^{2} + 53580549386883400 T^{3} + \)\(17\!\cdots\!86\)\( T^{4} + 53580549386883400 p^{11} T^{5} + 3829207972 p^{24} T^{6} + 99944 p^{33} T^{7} + p^{44} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 2306276 T + 7411071016186 T^{2} - 11446025290832742208 T^{3} + \)\(20\!\cdots\!35\)\( T^{4} - 11446025290832742208 p^{11} T^{5} + 7411071016186 p^{22} T^{6} - 2306276 p^{33} T^{7} + p^{44} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 3443816 T + 102603037682428 T^{2} + \)\(36\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!26\)\( T^{4} + \)\(36\!\cdots\!40\)\( p^{11} T^{5} + 102603037682428 p^{22} T^{6} + 3443816 p^{33} T^{7} + p^{44} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 4214548 T - 3406227372638 T^{2} - \)\(10\!\cdots\!24\)\( T^{3} + \)\(15\!\cdots\!59\)\( T^{4} - \)\(10\!\cdots\!24\)\( p^{11} T^{5} - 3406227372638 p^{22} T^{6} + 4214548 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 52691304 T + 4586182697931092 T^{2} - \)\(15\!\cdots\!84\)\( T^{3} + \)\(69\!\cdots\!10\)\( T^{4} - \)\(15\!\cdots\!84\)\( p^{11} T^{5} + 4586182697931092 p^{22} T^{6} - 52691304 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 217393304 T + 48076892013700300 T^{2} - \)\(67\!\cdots\!08\)\( T^{3} + \)\(90\!\cdots\!18\)\( T^{4} - \)\(67\!\cdots\!08\)\( p^{11} T^{5} + 48076892013700300 p^{22} T^{6} - 217393304 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 10526356 p T + 110894376843952498 T^{2} + \)\(21\!\cdots\!48\)\( T^{3} + \)\(42\!\cdots\!79\)\( T^{4} + \)\(21\!\cdots\!48\)\( p^{11} T^{5} + 110894376843952498 p^{22} T^{6} + 10526356 p^{34} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 273252872 T - 81057457446333812 T^{2} - \)\(33\!\cdots\!48\)\( T^{3} + \)\(69\!\cdots\!50\)\( T^{4} - \)\(33\!\cdots\!48\)\( p^{11} T^{5} - 81057457446333812 p^{22} T^{6} - 273252872 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 22069456 T + 796406960614594468 T^{2} + \)\(47\!\cdots\!48\)\( T^{3} + \)\(39\!\cdots\!14\)\( T^{4} + \)\(47\!\cdots\!48\)\( p^{11} T^{5} + 796406960614594468 p^{22} T^{6} + 22069456 p^{33} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 705091900 T + 3103662515659227250 T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!23\)\( T^{4} + \)\(18\!\cdots\!00\)\( p^{11} T^{5} + 3103662515659227250 p^{22} T^{6} + 705091900 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 187768360 T + 7352359892990648980 T^{2} - \)\(74\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!18\)\( T^{4} - \)\(74\!\cdots\!80\)\( p^{11} T^{5} + 7352359892990648980 p^{22} T^{6} - 187768360 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 6392224256 T + 955975989160217924 p T^{2} + \)\(18\!\cdots\!96\)\( T^{3} + \)\(76\!\cdots\!50\)\( T^{4} + \)\(18\!\cdots\!96\)\( p^{11} T^{5} + 955975989160217924 p^{23} T^{6} + 6392224256 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 36710008 T + 45110259195200964772 T^{2} + \)\(24\!\cdots\!04\)\( T^{3} + \)\(77\!\cdots\!34\)\( T^{4} + \)\(24\!\cdots\!04\)\( p^{11} T^{5} + 45110259195200964772 p^{22} T^{6} - 36710008 p^{33} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 11538870620 T + \)\(18\!\cdots\!46\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!71\)\( T^{4} - \)\(14\!\cdots\!80\)\( p^{11} T^{5} + \)\(18\!\cdots\!46\)\( p^{22} T^{6} - 11538870620 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 37721158484 T + \)\(95\!\cdots\!78\)\( T^{2} - \)\(15\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!51\)\( T^{4} - \)\(15\!\cdots\!60\)\( p^{11} T^{5} + \)\(95\!\cdots\!78\)\( p^{22} T^{6} - 37721158484 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 8211316688 T + \)\(77\!\cdots\!88\)\( T^{2} - \)\(53\!\cdots\!36\)\( T^{3} + \)\(25\!\cdots\!70\)\( T^{4} - \)\(53\!\cdots\!36\)\( p^{11} T^{5} + \)\(77\!\cdots\!88\)\( p^{22} T^{6} - 8211316688 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 5713413224 T + \)\(85\!\cdots\!72\)\( T^{2} - \)\(92\!\cdots\!84\)\( T^{3} + \)\(32\!\cdots\!30\)\( T^{4} - \)\(92\!\cdots\!84\)\( p^{11} T^{5} + \)\(85\!\cdots\!72\)\( p^{22} T^{6} - 5713413224 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 45026381600 T + \)\(17\!\cdots\!16\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!46\)\( T^{4} - \)\(23\!\cdots\!00\)\( p^{11} T^{5} + \)\(17\!\cdots\!16\)\( p^{22} T^{6} - 45026381600 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 104211315528 T + \)\(88\!\cdots\!40\)\( T^{2} + \)\(45\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!26\)\( T^{4} + \)\(45\!\cdots\!92\)\( p^{11} T^{5} + \)\(88\!\cdots\!40\)\( p^{22} T^{6} + 104211315528 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 111829609152 T + \)\(12\!\cdots\!52\)\( T^{2} - \)\(81\!\cdots\!04\)\( T^{3} + \)\(51\!\cdots\!14\)\( T^{4} - \)\(81\!\cdots\!04\)\( p^{11} T^{5} + \)\(12\!\cdots\!52\)\( p^{22} T^{6} - 111829609152 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 77104304804 T + \)\(13\!\cdots\!18\)\( T^{2} - \)\(68\!\cdots\!20\)\( p T^{3} + \)\(10\!\cdots\!91\)\( T^{4} - \)\(68\!\cdots\!20\)\( p^{12} T^{5} + \)\(13\!\cdots\!18\)\( p^{22} T^{6} - 77104304804 p^{33} T^{7} + p^{44} T^{8} \) | |
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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
−8.498739329450753880356305321412, −8.178390717632954702688501174151, −7.982710138626167600367425779875, −7.63345135121222104234367681150, −7.37398856515072045190563588436, −6.94419456417539568500345352162, −6.71830055181391585374619604429, −6.20493071358737972787246508087, −6.07790788461354987120833565243, −5.03443902978649334625114179160, −5.01115905592129563509100075422, −4.92919730503265247483494342473, −4.79160682077395355757916373410, −4.05810462395157098456561878800, −3.76041301036940798655882726546, −3.68336609170011967005490410973, −3.58571121415675529029236448451, −2.75930638512314840021188429477, −2.55155251650333866649655415343, −2.28305127468693985404942709665, −1.70146578954125649099124564759, −1.57562794030321949070765741846, −1.15032276629977750285029947991, −0.925628162238058979314186048494, −0.43420913448264102988323528447, 0.43420913448264102988323528447, 0.925628162238058979314186048494, 1.15032276629977750285029947991, 1.57562794030321949070765741846, 1.70146578954125649099124564759, 2.28305127468693985404942709665, 2.55155251650333866649655415343, 2.75930638512314840021188429477, 3.58571121415675529029236448451, 3.68336609170011967005490410973, 3.76041301036940798655882726546, 4.05810462395157098456561878800, 4.79160682077395355757916373410, 4.92919730503265247483494342473, 5.01115905592129563509100075422, 5.03443902978649334625114179160, 6.07790788461354987120833565243, 6.20493071358737972787246508087, 6.71830055181391585374619604429, 6.94419456417539568500345352162, 7.37398856515072045190563588436, 7.63345135121222104234367681150, 7.982710138626167600367425779875, 8.178390717632954702688501174151, 8.498739329450753880356305321412