Dirichlet series
| L(s) = 1 | + 64·2-s + 226·3-s + 2.56e3·4-s + 866·5-s + 1.44e4·6-s + 2.67e3·7-s + 8.19e4·8-s + 1.14e4·9-s + 5.54e4·10-s + 1.19e5·11-s + 5.78e5·12-s − 6.74e3·13-s + 1.70e5·14-s + 1.95e5·15-s + 2.29e6·16-s + 6.78e5·17-s + 7.34e5·18-s − 5.21e5·19-s + 2.21e6·20-s + 6.03e5·21-s + 7.63e6·22-s + 2.91e6·23-s + 1.85e7·24-s − 3.39e6·25-s − 4.31e5·26-s − 1.80e6·27-s + 6.83e6·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.61·3-s + 5·4-s + 0.619·5-s + 4.55·6-s + 0.420·7-s + 7.07·8-s + 0.582·9-s + 1.75·10-s + 2.45·11-s + 8.05·12-s − 0.0655·13-s + 1.18·14-s + 0.998·15-s + 35/4·16-s + 1.97·17-s + 1.64·18-s − 0.917·19-s + 3.09·20-s + 0.677·21-s + 6.94·22-s + 2.16·23-s + 11.3·24-s − 1.73·25-s − 0.185·26-s − 0.652·27-s + 2.10·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2085136\) = \(2^{4} \cdot 19^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(146718.\) |
| Root analytic conductor: | \(4.42395\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2085136,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(128.4563181\) |
| \(L(\frac12)\) | \(\approx\) | \(128.4563181\) |
| \(L(\frac{11}{2})\) | 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 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 19 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 226 T + 39601 T^{2} - 694 p^{8} T^{3} + 49573448 p^{2} T^{4} - 694 p^{17} T^{5} + 39601 p^{18} T^{6} - 226 p^{27} T^{7} + p^{36} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 866 T + 829369 p T^{2} - 190343618 p^{2} T^{3} + 83373323792 p^{3} T^{4} - 190343618 p^{11} T^{5} + 829369 p^{19} T^{6} - 866 p^{27} T^{7} + p^{36} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 - 2670 T + 99717568 T^{2} - 29542339224 p T^{3} + 113825580686553 p^{2} T^{4} - 29542339224 p^{10} T^{5} + 99717568 p^{18} T^{6} - 2670 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 119234 T + 11711266961 T^{2} - 777508728576194 T^{3} + 44417401991215600432 T^{4} - 777508728576194 p^{9} T^{5} + 11711266961 p^{18} T^{6} - 119234 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 6748 T + 20740879831 T^{2} - 83767778884340 p T^{3} + \)\(20\!\cdots\!44\)\( T^{4} - 83767778884340 p^{10} T^{5} + 20740879831 p^{18} T^{6} + 6748 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 678624 T + 301210589426 T^{2} - 139032189424965312 T^{3} + \)\(60\!\cdots\!35\)\( T^{4} - 139032189424965312 p^{9} T^{5} + 301210589426 p^{18} T^{6} - 678624 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 2911868 T + 7020626597375 T^{2} - 565511821273945876 p T^{3} + \)\(18\!\cdots\!24\)\( T^{4} - 565511821273945876 p^{10} T^{5} + 7020626597375 p^{18} T^{6} - 2911868 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 8291104 T + 65007930953507 T^{2} - \)\(28\!\cdots\!48\)\( T^{3} + \)\(13\!\cdots\!44\)\( T^{4} - \)\(28\!\cdots\!48\)\( p^{9} T^{5} + 65007930953507 p^{18} T^{6} - 8291104 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 3445468 T + 76079895372940 T^{2} - \)\(24\!\cdots\!04\)\( T^{3} + \)\(27\!\cdots\!42\)\( T^{4} - \)\(24\!\cdots\!04\)\( p^{9} T^{5} + 76079895372940 p^{18} T^{6} - 3445468 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 1005524 T + 61996920196900 T^{2} - \)\(10\!\cdots\!84\)\( T^{3} - \)\(87\!\cdots\!06\)\( T^{4} - \)\(10\!\cdots\!84\)\( p^{9} T^{5} + 61996920196900 p^{18} T^{6} + 1005524 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 8514124 T + 576067992126680 T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!26\)\( T^{4} - \)\(62\!\cdots\!80\)\( p^{9} T^{5} + 576067992126680 p^{18} T^{6} - 8514124 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 13900726 T + 1263032063045605 T^{2} - \)\(49\!\cdots\!66\)\( T^{3} + \)\(69\!\cdots\!60\)\( T^{4} - \)\(49\!\cdots\!66\)\( p^{9} T^{5} + 1263032063045605 p^{18} T^{6} - 13900726 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 36334954 T + 1333040393862797 T^{2} + \)\(22\!\cdots\!54\)\( T^{3} + \)\(16\!\cdots\!20\)\( T^{4} + \)\(22\!\cdots\!54\)\( p^{9} T^{5} + 1333040393862797 p^{18} T^{6} + 36334954 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 113969356 T + 10662975352067663 T^{2} + \)\(54\!\cdots\!12\)\( T^{3} + \)\(35\!\cdots\!72\)\( T^{4} + \)\(54\!\cdots\!12\)\( p^{9} T^{5} + 10662975352067663 p^{18} T^{6} + 113969356 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 396773766 T + 84015153907615097 T^{2} + \)\(11\!\cdots\!22\)\( T^{3} + \)\(12\!\cdots\!24\)\( T^{4} + \)\(11\!\cdots\!22\)\( p^{9} T^{5} + 84015153907615097 p^{18} T^{6} + 396773766 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 298192066 T + 68250132645236185 T^{2} + \)\(10\!\cdots\!50\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(10\!\cdots\!50\)\( p^{9} T^{5} + 68250132645236185 p^{18} T^{6} + 298192066 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 113551722 T + 50137220305978117 T^{2} - \)\(13\!\cdots\!94\)\( T^{3} + \)\(78\!\cdots\!32\)\( T^{4} - \)\(13\!\cdots\!94\)\( p^{9} T^{5} + 50137220305978117 p^{18} T^{6} + 113551722 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 4659620 T + 66622804019153564 T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!34\)\( T^{4} + \)\(12\!\cdots\!88\)\( p^{9} T^{5} + 66622804019153564 p^{18} T^{6} - 4659620 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 136198452 T + 159202022806579642 T^{2} - \)\(18\!\cdots\!28\)\( T^{3} + \)\(13\!\cdots\!11\)\( T^{4} - \)\(18\!\cdots\!28\)\( p^{9} T^{5} + 159202022806579642 p^{18} T^{6} - 136198452 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 67255424 T + 466792363237010728 T^{2} - \)\(23\!\cdots\!08\)\( T^{3} + \)\(83\!\cdots\!42\)\( T^{4} - \)\(23\!\cdots\!08\)\( p^{9} T^{5} + 466792363237010728 p^{18} T^{6} - 67255424 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 1376505216 T + 1405885246005088616 T^{2} - \)\(90\!\cdots\!24\)\( T^{3} + \)\(46\!\cdots\!02\)\( T^{4} - \)\(90\!\cdots\!24\)\( p^{9} T^{5} + 1405885246005088616 p^{18} T^{6} - 1376505216 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 1557211260 T + 2036231523650812436 T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!86\)\( T^{4} - \)\(16\!\cdots\!20\)\( p^{9} T^{5} + 2036231523650812436 p^{18} T^{6} - 1557211260 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 975818188 T + 3235606984725522088 T^{2} - \)\(22\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!30\)\( T^{4} - \)\(22\!\cdots\!80\)\( p^{9} T^{5} + 3235606984725522088 p^{18} T^{6} - 975818188 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−10.30632513617105200964319312286, −9.663900724882127217429430551842, −9.382731960742098120132059111881, −9.219743575779174238093632426688, −8.878791392755431473201944960245, −8.020882403097535426032690016507, −7.951545027585531099812787873668, −7.82488115513257389497274913201, −7.24892077966604689948191128989, −6.52341566069368986167758514889, −6.37902282996600690107669156367, −6.21913989977530491476035016192, −5.98506718188984797225884385318, −5.06134790947131822950083339280, −4.96771480734055571258555767301, −4.40350550170063210126729924545, −4.34721761367068607904431943602, −3.43533471742143095075850093057, −3.37114172157925745113457582998, −2.99996472236985578391572855424, −2.87261425884292986877075981537, −1.92609214215660153127208254442, −1.66034081094288870575595064779, −1.44169155849399242590803488281, −0.78405756366905876824752866698, 0.78405756366905876824752866698, 1.44169155849399242590803488281, 1.66034081094288870575595064779, 1.92609214215660153127208254442, 2.87261425884292986877075981537, 2.99996472236985578391572855424, 3.37114172157925745113457582998, 3.43533471742143095075850093057, 4.34721761367068607904431943602, 4.40350550170063210126729924545, 4.96771480734055571258555767301, 5.06134790947131822950083339280, 5.98506718188984797225884385318, 6.21913989977530491476035016192, 6.37902282996600690107669156367, 6.52341566069368986167758514889, 7.24892077966604689948191128989, 7.82488115513257389497274913201, 7.951545027585531099812787873668, 8.020882403097535426032690016507, 8.878791392755431473201944960245, 9.219743575779174238093632426688, 9.382731960742098120132059111881, 9.663900724882127217429430551842, 10.30632513617105200964319312286