Dirichlet series
| L(s) = 1 | − 41·2-s − 324·3-s + 67·4-s + 2.50e3·5-s + 1.32e4·6-s + 9.60e3·7-s + 1.74e4·8-s + 6.56e4·9-s − 1.02e5·10-s − 3.28e4·11-s − 2.17e4·12-s − 1.33e5·13-s − 3.93e5·14-s − 8.10e5·15-s − 2.05e5·16-s + 3.34e5·17-s − 2.69e6·18-s + 7.20e4·19-s + 1.67e5·20-s − 3.11e6·21-s + 1.34e6·22-s + 4.74e4·23-s − 5.64e6·24-s + 3.90e6·25-s + 5.48e6·26-s − 1.06e7·27-s + 6.43e5·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 2.30·3-s + 0.130·4-s + 1.78·5-s + 4.18·6-s + 1.51·7-s + 1.50·8-s + 10/3·9-s − 3.24·10-s − 0.676·11-s − 0.302·12-s − 1.30·13-s − 2.73·14-s − 4.13·15-s − 0.784·16-s + 0.970·17-s − 6.03·18-s + 0.126·19-s + 0.234·20-s − 3.49·21-s + 1.22·22-s + 0.0353·23-s − 3.47·24-s + 2·25-s + 2.35·26-s − 3.84·27-s + 0.197·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(3^{4} \cdot 5^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.55277\times 10^{6}\) |
| Root analytic conductor: | \(7.35382\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | 3 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 41 T + 807 p T^{2} + 5749 p^{3} T^{3} + 2477 p^{9} T^{4} + 5749 p^{12} T^{5} + 807 p^{19} T^{6} + 41 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 32854 T + 4407241540 T^{2} + 279186576359286 T^{3} + 11133742738243632614 T^{4} + 279186576359286 p^{9} T^{5} + 4407241540 p^{18} T^{6} + 32854 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 133882 T + 24616076456 T^{2} + 1870759342691086 T^{3} + \)\(24\!\cdots\!34\)\( T^{4} + 1870759342691086 p^{9} T^{5} + 24616076456 p^{18} T^{6} + 133882 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 334344 T + 253836269084 T^{2} - 21001627928195768 T^{3} + \)\(22\!\cdots\!14\)\( T^{4} - 21001627928195768 p^{9} T^{5} + 253836269084 p^{18} T^{6} - 334344 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 72046 T + 113280964052 T^{2} - 64317643040518238 T^{3} + \)\(10\!\cdots\!90\)\( T^{4} - 64317643040518238 p^{9} T^{5} + 113280964052 p^{18} T^{6} - 72046 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 47460 T + 5151065185052 T^{2} - 334917001921405940 T^{3} + \)\(12\!\cdots\!14\)\( T^{4} - 334917001921405940 p^{9} T^{5} + 5151065185052 p^{18} T^{6} - 47460 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 7010312 T + 61187331697900 T^{2} - \)\(24\!\cdots\!12\)\( T^{3} + \)\(12\!\cdots\!14\)\( T^{4} - \)\(24\!\cdots\!12\)\( p^{9} T^{5} + 61187331697900 p^{18} T^{6} - 7010312 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 3711690 T + 69320238083404 T^{2} + \)\(22\!\cdots\!70\)\( T^{3} + \)\(26\!\cdots\!86\)\( T^{4} + \)\(22\!\cdots\!70\)\( p^{9} T^{5} + 69320238083404 p^{18} T^{6} + 3711690 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 5222716 T + 121113162182324 T^{2} + 81701486034883838188 T^{3} + \)\(16\!\cdots\!14\)\( T^{4} + 81701486034883838188 p^{9} T^{5} + 121113162182324 p^{18} T^{6} - 5222716 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 5689292 T + 156721503522468 T^{2} + \)\(53\!\cdots\!04\)\( T^{3} + \)\(89\!\cdots\!70\)\( T^{4} + \)\(53\!\cdots\!04\)\( p^{9} T^{5} + 156721503522468 p^{18} T^{6} + 5689292 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 71286384 T + 3216287652243148 T^{2} + \)\(95\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!66\)\( T^{4} + \)\(95\!\cdots\!40\)\( p^{9} T^{5} + 3216287652243148 p^{18} T^{6} + 71286384 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 100832160 T + 6330075229629948 T^{2} + \)\(27\!\cdots\!60\)\( T^{3} + \)\(99\!\cdots\!54\)\( T^{4} + \)\(27\!\cdots\!60\)\( p^{9} T^{5} + 6330075229629948 p^{18} T^{6} + 100832160 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 149250454 T + 20132596617058208 T^{2} + \)\(15\!\cdots\!90\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} + \)\(15\!\cdots\!90\)\( p^{9} T^{5} + 20132596617058208 p^{18} T^{6} + 149250454 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 24779068 T + 19883054657391420 T^{2} + \)\(17\!\cdots\!72\)\( T^{3} + \)\(18\!\cdots\!34\)\( T^{4} + \)\(17\!\cdots\!72\)\( p^{9} T^{5} + 19883054657391420 p^{18} T^{6} - 24779068 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 301336048 T + 66672244999315708 T^{2} + \)\(99\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} + \)\(99\!\cdots\!36\)\( p^{9} T^{5} + 66672244999315708 p^{18} T^{6} + 301336048 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 10978016 T + 64766334909335404 T^{2} - \)\(46\!\cdots\!28\)\( T^{3} + \)\(21\!\cdots\!34\)\( T^{4} - \)\(46\!\cdots\!28\)\( p^{9} T^{5} + 64766334909335404 p^{18} T^{6} + 10978016 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 507837170 T + 194326881986686444 T^{2} + \)\(77\!\cdots\!10\)\( p T^{3} + \)\(13\!\cdots\!06\)\( T^{4} + \)\(77\!\cdots\!10\)\( p^{10} T^{5} + 194326881986686444 p^{18} T^{6} + 507837170 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 158796526 T + 198583211037575048 T^{2} + \)\(23\!\cdots\!90\)\( T^{3} + \)\(16\!\cdots\!86\)\( T^{4} + \)\(23\!\cdots\!90\)\( p^{9} T^{5} + 198583211037575048 p^{18} T^{6} + 158796526 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 36676584 T + 230034315686625852 T^{2} + \)\(44\!\cdots\!92\)\( T^{3} + \)\(29\!\cdots\!90\)\( T^{4} + \)\(44\!\cdots\!92\)\( p^{9} T^{5} + 230034315686625852 p^{18} T^{6} + 36676584 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 201009048 T + 682349377451611596 T^{2} - \)\(10\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!74\)\( T^{4} - \)\(10\!\cdots\!24\)\( p^{9} T^{5} + 682349377451611596 p^{18} T^{6} - 201009048 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 161323024 T + 243639186907299372 T^{2} + \)\(51\!\cdots\!48\)\( T^{3} + \)\(15\!\cdots\!50\)\( T^{4} + \)\(51\!\cdots\!48\)\( p^{9} T^{5} + 243639186907299372 p^{18} T^{6} - 161323024 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1913030882 T + 3698361872070074272 T^{2} + \)\(40\!\cdots\!50\)\( T^{3} + \)\(44\!\cdots\!46\)\( T^{4} + \)\(40\!\cdots\!50\)\( p^{9} T^{5} + 3698361872070074272 p^{18} T^{6} + 1913030882 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−9.223764759778297997129570157734, −8.458496756352975908274367314594, −8.233437793575933337705899990271, −8.154213744138152227385198186513, −7.985441439536605435004717105584, −7.50159938193639292012110761261, −7.05065131223665426914380586676, −6.79363307488604606035657464908, −6.49077975974395772403825897282, −6.30167108342216378586000453329, −5.72133114470839120995784560999, −5.60506691508368722108026935891, −5.36426585449451528076858850328, −4.79361672822217341598462138901, −4.76920781973455673580245982478, −4.59815704617937865968703787563, −4.56603411669057536681134032591, −3.37467852650433420257686078399, −3.25048879530263207538144554077, −2.56314904915299764996316143633, −2.29880992150977619848151006470, −1.54466353827995654564639158533, −1.39886050136172621474309389817, −1.22948254764012627743126428457, −1.14978793354320743683906337796, 0, 0, 0, 0, 1.14978793354320743683906337796, 1.22948254764012627743126428457, 1.39886050136172621474309389817, 1.54466353827995654564639158533, 2.29880992150977619848151006470, 2.56314904915299764996316143633, 3.25048879530263207538144554077, 3.37467852650433420257686078399, 4.56603411669057536681134032591, 4.59815704617937865968703787563, 4.76920781973455673580245982478, 4.79361672822217341598462138901, 5.36426585449451528076858850328, 5.60506691508368722108026935891, 5.72133114470839120995784560999, 6.30167108342216378586000453329, 6.49077975974395772403825897282, 6.79363307488604606035657464908, 7.05065131223665426914380586676, 7.50159938193639292012110761261, 7.985441439536605435004717105584, 8.154213744138152227385198186513, 8.233437793575933337705899990271, 8.458496756352975908274367314594, 9.223764759778297997129570157734