Dirichlet series
| L(s) = 1 | − 16·2-s − 68·3-s − 64·4-s − 56·5-s + 1.08e3·6-s − 1.15e3·7-s + 1.08e3·8-s − 4.25e3·9-s + 896·10-s − 1.31e3·11-s + 4.35e3·12-s − 1.96e4·13-s + 1.84e4·14-s + 3.80e3·15-s + 1.73e4·16-s − 5.00e3·17-s + 6.80e4·18-s − 3.83e4·19-s + 3.58e3·20-s + 7.86e4·21-s + 2.10e4·22-s + 6.08e4·23-s − 7.39e4·24-s − 1.66e5·25-s + 3.14e5·26-s + 4.77e5·27-s + 7.39e4·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.45·3-s − 1/2·4-s − 0.200·5-s + 2.05·6-s − 1.27·7-s + 0.751·8-s − 1.94·9-s + 0.283·10-s − 0.298·11-s + 0.727·12-s − 2.48·13-s + 1.80·14-s + 0.291·15-s + 1.05·16-s − 0.246·17-s + 2.75·18-s − 1.28·19-s + 0.100·20-s + 1.85·21-s + 0.422·22-s + 1.04·23-s − 1.09·24-s − 2.12·25-s + 3.51·26-s + 4.66·27-s + 0.636·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 6436343 ^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6436343 ^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(6436343\) = \(23^{5}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(19146.5\) |
| Root analytic conductor: | \(2.68045\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(5\) |
| Selberg data: | \((10,\ 6436343,\ (\ :7/2, 7/2, 7/2, 7/2, 7/2),\ -1)\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | 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 | 23 | $C_1$ | \( ( 1 - p^{3} T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 + p^{4} T + 5 p^{6} T^{2} + 79 p^{6} T^{3} + 4165 p^{4} T^{4} + 12201 p^{6} T^{5} + 4165 p^{11} T^{6} + 79 p^{20} T^{7} + 5 p^{27} T^{8} + p^{32} T^{9} + p^{35} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 + 68 T + 2959 p T^{2} + 46180 p^{2} T^{3} + 1213315 p^{3} T^{4} + 14624336 p^{4} T^{5} + 1213315 p^{10} T^{6} + 46180 p^{16} T^{7} + 2959 p^{22} T^{8} + 68 p^{28} T^{9} + p^{35} T^{10} \) | |
| 5 | $C_2 \wr S_5$ | \( 1 + 56 T + 169401 T^{2} + 26920592 T^{3} + 22521136946 T^{4} + 372614133488 p T^{5} + 22521136946 p^{7} T^{6} + 26920592 p^{14} T^{7} + 169401 p^{21} T^{8} + 56 p^{28} T^{9} + p^{35} T^{10} \) | |
| 7 | $C_2 \wr S_5$ | \( 1 + 1156 T + 76115 p^{2} T^{2} + 3696970176 T^{3} + 833754193462 p T^{4} + 4522847850159096 T^{5} + 833754193462 p^{8} T^{6} + 3696970176 p^{14} T^{7} + 76115 p^{23} T^{8} + 1156 p^{28} T^{9} + p^{35} T^{10} \) | |
| 11 | $C_2 \wr S_5$ | \( 1 + 1318 T + 37730119 T^{2} - 23938903864 T^{3} + 254033555505954 T^{4} - 1932207002605083484 T^{5} + 254033555505954 p^{7} T^{6} - 23938903864 p^{14} T^{7} + 37730119 p^{21} T^{8} + 1318 p^{28} T^{9} + p^{35} T^{10} \) | |
| 13 | $C_2 \wr S_5$ | \( 1 + 19662 T + 29142627 p T^{2} + 4338212129016 T^{3} + 48606675949638613 T^{4} + \)\(38\!\cdots\!02\)\( T^{5} + 48606675949638613 p^{7} T^{6} + 4338212129016 p^{14} T^{7} + 29142627 p^{22} T^{8} + 19662 p^{28} T^{9} + p^{35} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 + 5002 T + 1861917557 T^{2} + 8305744532632 T^{3} + 1451605340654732458 T^{4} + \)\(51\!\cdots\!56\)\( T^{5} + 1451605340654732458 p^{7} T^{6} + 8305744532632 p^{14} T^{7} + 1861917557 p^{21} T^{8} + 5002 p^{28} T^{9} + p^{35} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 + 38314 T + 2915565487 T^{2} + 75468157912264 T^{3} + 3701899105425815698 T^{4} + \)\(74\!\cdots\!64\)\( T^{5} + 3701899105425815698 p^{7} T^{6} + 75468157912264 p^{14} T^{7} + 2915565487 p^{21} T^{8} + 38314 p^{28} T^{9} + p^{35} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 + 150634 T + 54700931303 T^{2} + 8850823815865800 T^{3} + 54468911769131861849 p T^{4} + \)\(21\!\cdots\!42\)\( T^{5} + 54468911769131861849 p^{8} T^{6} + 8850823815865800 p^{14} T^{7} + 54700931303 p^{21} T^{8} + 150634 p^{28} T^{9} + p^{35} T^{10} \) | |
| 31 | $C_2 \wr S_5$ | \( 1 + 179940 T + 74755448001 T^{2} + 10182233369835364 T^{3} + \)\(29\!\cdots\!21\)\( T^{4} + \)\(30\!\cdots\!28\)\( T^{5} + \)\(29\!\cdots\!21\)\( p^{7} T^{6} + 10182233369835364 p^{14} T^{7} + 74755448001 p^{21} T^{8} + 179940 p^{28} T^{9} + p^{35} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 + 752672 T + 411187107993 T^{2} + 143229152414097936 T^{3} + \)\(41\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!44\)\( T^{5} + \)\(41\!\cdots\!06\)\( p^{7} T^{6} + 143229152414097936 p^{14} T^{7} + 411187107993 p^{21} T^{8} + 752672 p^{28} T^{9} + p^{35} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 + 1192910 T + 1234839405067 T^{2} + 742378770518958104 T^{3} + \)\(43\!\cdots\!85\)\( T^{4} + \)\(18\!\cdots\!98\)\( T^{5} + \)\(43\!\cdots\!85\)\( p^{7} T^{6} + 742378770518958104 p^{14} T^{7} + 1234839405067 p^{21} T^{8} + 1192910 p^{28} T^{9} + p^{35} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 + 932646 T + 1230007277807 T^{2} + 915631012777150664 T^{3} + \)\(65\!\cdots\!02\)\( T^{4} + \)\(35\!\cdots\!40\)\( T^{5} + \)\(65\!\cdots\!02\)\( p^{7} T^{6} + 915631012777150664 p^{14} T^{7} + 1230007277807 p^{21} T^{8} + 932646 p^{28} T^{9} + p^{35} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 + 1008460 T + 43886500911 p T^{2} + 1744029648899142180 T^{3} + \)\(18\!\cdots\!13\)\( T^{4} + \)\(12\!\cdots\!12\)\( T^{5} + \)\(18\!\cdots\!13\)\( p^{7} T^{6} + 1744029648899142180 p^{14} T^{7} + 43886500911 p^{22} T^{8} + 1008460 p^{28} T^{9} + p^{35} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 - 897104 T + 3419538434473 T^{2} - 3061558251917842656 T^{3} + \)\(62\!\cdots\!42\)\( T^{4} - \)\(45\!\cdots\!76\)\( T^{5} + \)\(62\!\cdots\!42\)\( p^{7} T^{6} - 3061558251917842656 p^{14} T^{7} + 3419538434473 p^{21} T^{8} - 897104 p^{28} T^{9} + p^{35} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 - 1020972 T + 2018890240735 T^{2} - 6952810938395061648 T^{3} + \)\(94\!\cdots\!18\)\( T^{4} - \)\(81\!\cdots\!92\)\( T^{5} + \)\(94\!\cdots\!18\)\( p^{7} T^{6} - 6952810938395061648 p^{14} T^{7} + 2018890240735 p^{21} T^{8} - 1020972 p^{28} T^{9} + p^{35} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 + 2758364 T + 12045320168321 T^{2} + 25655779189423888816 T^{3} + \)\(68\!\cdots\!54\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(68\!\cdots\!54\)\( p^{7} T^{6} + 25655779189423888816 p^{14} T^{7} + 12045320168321 p^{21} T^{8} + 2758364 p^{28} T^{9} + p^{35} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 + 1523138 T + 20873284382319 T^{2} + 19775624408901921432 T^{3} + \)\(20\!\cdots\!06\)\( T^{4} + \)\(14\!\cdots\!84\)\( T^{5} + \)\(20\!\cdots\!06\)\( p^{7} T^{6} + 19775624408901921432 p^{14} T^{7} + 20873284382319 p^{21} T^{8} + 1523138 p^{28} T^{9} + p^{35} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 - 3044884 T + 29925590131929 T^{2} - 73285084748881262812 T^{3} + \)\(39\!\cdots\!41\)\( T^{4} - \)\(84\!\cdots\!96\)\( T^{5} + \)\(39\!\cdots\!41\)\( p^{7} T^{6} - 73285084748881262812 p^{14} T^{7} + 29925590131929 p^{21} T^{8} - 3044884 p^{28} T^{9} + p^{35} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 + 8872022 T + 70542647809643 T^{2} + \)\(35\!\cdots\!96\)\( T^{3} + \)\(16\!\cdots\!97\)\( T^{4} + \)\(56\!\cdots\!34\)\( T^{5} + \)\(16\!\cdots\!97\)\( p^{7} T^{6} + \)\(35\!\cdots\!96\)\( p^{14} T^{7} + 70542647809643 p^{21} T^{8} + 8872022 p^{28} T^{9} + p^{35} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 + 4437540 T + 54847911621507 T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!54\)\( T^{4} + \)\(40\!\cdots\!68\)\( T^{5} + \)\(15\!\cdots\!54\)\( p^{7} T^{6} + \)\(16\!\cdots\!80\)\( p^{14} T^{7} + 54847911621507 p^{21} T^{8} + 4437540 p^{28} T^{9} + p^{35} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 + 4637362 T + 54681653954327 T^{2} + 2629382495809588888 p T^{3} + \)\(22\!\cdots\!42\)\( T^{4} + \)\(75\!\cdots\!32\)\( T^{5} + \)\(22\!\cdots\!42\)\( p^{7} T^{6} + 2629382495809588888 p^{15} T^{7} + 54681653954327 p^{21} T^{8} + 4637362 p^{28} T^{9} + p^{35} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 - 6381402 T + 163313414269101 T^{2} - \)\(74\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!74\)\( T^{4} - \)\(41\!\cdots\!24\)\( T^{5} + \)\(11\!\cdots\!74\)\( p^{7} T^{6} - \)\(74\!\cdots\!36\)\( p^{14} T^{7} + 163313414269101 p^{21} T^{8} - 6381402 p^{28} T^{9} + p^{35} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 + 6432034 T + 47522772361861 T^{2} + \)\(90\!\cdots\!32\)\( T^{3} + \)\(81\!\cdots\!18\)\( T^{4} + \)\(70\!\cdots\!08\)\( T^{5} + \)\(81\!\cdots\!18\)\( p^{7} T^{6} + \)\(90\!\cdots\!32\)\( p^{14} T^{7} + 47522772361861 p^{21} T^{8} + 6432034 p^{28} T^{9} + p^{35} T^{10} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.97635536467789903785300662041, −10.04750736737213245528878393955, −10.03932429343437680870002717677, −10.02734348157945340892696587469, −9.855154061845855426011124582736, −9.109416926187915999903194812307, −9.019838499030636337384547371009, −8.723456960651654356925819833817, −8.501134136330902748433768056869, −8.429267266603469236682227211331, −7.63694394624244850082204076734, −7.57976667560799655325124370504, −6.97288605279183391065060784530, −6.44580077963073206267366293860, −6.39885422662374251451656667521, −6.00569493245155702980508637688, −5.34241543156507488387179929533, −5.31965024070023853876465709875, −4.97311341772698168062983402708, −4.64307867786491467065333592975, −3.60757870256352111127078603481, −3.16804945304163144511176811500, −3.15282368904463550948938733752, −2.24183350314951070509321999895, −1.68412559612227439245546457097, 0, 0, 0, 0, 0, 1.68412559612227439245546457097, 2.24183350314951070509321999895, 3.15282368904463550948938733752, 3.16804945304163144511176811500, 3.60757870256352111127078603481, 4.64307867786491467065333592975, 4.97311341772698168062983402708, 5.31965024070023853876465709875, 5.34241543156507488387179929533, 6.00569493245155702980508637688, 6.39885422662374251451656667521, 6.44580077963073206267366293860, 6.97288605279183391065060784530, 7.57976667560799655325124370504, 7.63694394624244850082204076734, 8.429267266603469236682227211331, 8.501134136330902748433768056869, 8.723456960651654356925819833817, 9.019838499030636337384547371009, 9.109416926187915999903194812307, 9.855154061845855426011124582736, 10.02734348157945340892696587469, 10.03932429343437680870002717677, 10.04750736737213245528878393955, 10.97635536467789903785300662041