Properties

 Degree $12$ Conductor $15625$ Sign $1$ Motivic weight $8$ Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·2-s − 72·3-s + 2·4-s + 220·5-s + 144·6-s − 2.35e3·7-s − 2.91e3·8-s + 2.59e3·9-s − 440·10-s + 2.31e4·11-s − 144·12-s − 1.19e5·13-s + 4.70e3·14-s − 1.58e4·15-s + 6.27e4·16-s − 2.65e5·17-s − 5.18e3·18-s + 440·20-s + 1.69e5·21-s − 4.63e4·22-s + 2.88e4·23-s + 2.09e5·24-s − 1.45e5·25-s + 2.38e5·26-s − 8.89e4·27-s − 4.70e3·28-s + 3.16e4·30-s + ⋯
 L(s)  = 1 − 1/8·2-s − 8/9·3-s + 0.00781·4-s + 0.351·5-s + 1/9·6-s − 0.979·7-s − 0.710·8-s + 0.395·9-s − 0.0439·10-s + 1.58·11-s − 0.00694·12-s − 4.17·13-s + 6/49·14-s − 0.312·15-s + 0.957·16-s − 3.17·17-s − 0.0493·18-s + 0.00274·20-s + 0.870·21-s − 0.198·22-s + 0.103·23-s + 0.631·24-s − 0.373·25-s + 0.521·26-s − 0.167·27-s − 0.00765·28-s + 0.0391·30-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s+4)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$12$$ Conductor: $$15625$$    =    $$5^{6}$$ Sign: $1$ Motivic weight: $$8$$ Character: induced by $\chi_{5} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 15625,\ (\ :[4]^{6}),\ 1)$$

Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.0185895$$ $$L(\frac12)$$ $$\approx$$ $$0.0185895$$ $$L(5)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 - 44 p T + 311 p^{4} T^{2} - 2552 p^{7} T^{3} + 311 p^{12} T^{4} - 44 p^{17} T^{5} + p^{24} T^{6}$$
good2 $$1 + p T + p T^{2} + 91 p^{5} T^{3} - 799 p^{6} T^{4} - 8873 p^{7} T^{5} + 16177 p^{7} T^{6} - 8873 p^{15} T^{7} - 799 p^{22} T^{8} + 91 p^{29} T^{9} + p^{33} T^{10} + p^{41} T^{11} + p^{48} T^{12}$$
3 $$1 + 8 p^{2} T + 32 p^{4} T^{2} + 3296 p^{3} T^{3} - 4782569 p^{2} T^{4} - 45184904 p^{4} T^{5} - 203005216 p^{6} T^{6} - 45184904 p^{12} T^{7} - 4782569 p^{18} T^{8} + 3296 p^{27} T^{9} + 32 p^{36} T^{10} + 8 p^{42} T^{11} + p^{48} T^{12}$$
7 $$1 + 48 p^{2} T + 1152 p^{4} T^{2} - 6884936 p^{3} T^{3} + 2619602799 p^{4} T^{4} + 7645236875928 p^{5} T^{5} + 2444629423375904 p^{6} T^{6} + 7645236875928 p^{13} T^{7} + 2619602799 p^{20} T^{8} - 6884936 p^{27} T^{9} + 1152 p^{36} T^{10} + 48 p^{42} T^{11} + p^{48} T^{12}$$
11 $$( 1 - 11596 T + 493098215 T^{2} - 5106545516920 T^{3} + 493098215 p^{8} T^{4} - 11596 p^{16} T^{5} + p^{24} T^{6} )^{2}$$
13 $$1 + 119142 T + 7097408082 T^{2} + 332686420223782 T^{3} + 13680551086291514559 T^{4} +$$$$46\!\cdots\!56$$$$T^{5} +$$$$13\!\cdots\!76$$$$T^{6} +$$$$46\!\cdots\!56$$$$p^{8} T^{7} + 13680551086291514559 p^{16} T^{8} + 332686420223782 p^{24} T^{9} + 7097408082 p^{32} T^{10} + 119142 p^{40} T^{11} + p^{48} T^{12}$$
17 $$1 + 265502 T + 35245656002 T^{2} + 3505767378301982 T^{3} +$$$$37\!\cdots\!19$$$$T^{4} +$$$$40\!\cdots\!76$$$$T^{5} +$$$$37\!\cdots\!76$$$$T^{6} +$$$$40\!\cdots\!76$$$$p^{8} T^{7} +$$$$37\!\cdots\!19$$$$p^{16} T^{8} + 3505767378301982 p^{24} T^{9} + 35245656002 p^{32} T^{10} + 265502 p^{40} T^{11} + p^{48} T^{12}$$
19 $$1 - 66391003446 T^{2} +$$$$21\!\cdots\!15$$$$T^{4} -$$$$42\!\cdots\!20$$$$T^{6} +$$$$21\!\cdots\!15$$$$p^{16} T^{8} - 66391003446 p^{32} T^{10} + p^{48} T^{12}$$
23 $$1 - 1256 p T + 788768 p^{2} T^{2} - 2207314762520128 T^{3} +$$$$86\!\cdots\!39$$$$T^{4} -$$$$83\!\cdots\!64$$$$T^{5} +$$$$12\!\cdots\!16$$$$T^{6} -$$$$83\!\cdots\!64$$$$p^{8} T^{7} +$$$$86\!\cdots\!39$$$$p^{16} T^{8} - 2207314762520128 p^{24} T^{9} + 788768 p^{34} T^{10} - 1256 p^{41} T^{11} + p^{48} T^{12}$$
29 $$1 - 1726260912966 T^{2} +$$$$13\!\cdots\!15$$$$T^{4} -$$$$77\!\cdots\!20$$$$T^{6} +$$$$13\!\cdots\!15$$$$p^{16} T^{8} - 1726260912966 p^{32} T^{10} + p^{48} T^{12}$$
31 $$( 1 + 373824 T + 1438821265815 T^{2} + 103488660558742480 T^{3} + 1438821265815 p^{8} T^{4} + 373824 p^{16} T^{5} + p^{24} T^{6} )^{2}$$
37 $$1 + 454002 T + 103058908002 T^{2} + 1591257258997412242 T^{3} +$$$$36\!\cdots\!59$$$$T^{4} +$$$$11\!\cdots\!36$$$$T^{5} +$$$$25\!\cdots\!36$$$$T^{6} +$$$$11\!\cdots\!36$$$$p^{8} T^{7} +$$$$36\!\cdots\!59$$$$p^{16} T^{8} + 1591257258997412242 p^{24} T^{9} + 103058908002 p^{32} T^{10} + 454002 p^{40} T^{11} + p^{48} T^{12}$$
41 $$( 1 - 1244716 T + 19779856453415 T^{2} - 17348876621060070520 T^{3} + 19779856453415 p^{8} T^{4} - 1244716 p^{16} T^{5} + p^{24} T^{6} )^{2}$$
43 $$1 - 792648 T + 314145425952 T^{2} - 6701894073526462448 T^{3} +$$$$27\!\cdots\!99$$$$T^{4} -$$$$18\!\cdots\!04$$$$T^{5} +$$$$86\!\cdots\!96$$$$T^{6} -$$$$18\!\cdots\!04$$$$p^{8} T^{7} +$$$$27\!\cdots\!99$$$$p^{16} T^{8} - 6701894073526462448 p^{24} T^{9} + 314145425952 p^{32} T^{10} - 792648 p^{40} T^{11} + p^{48} T^{12}$$
47 $$1 + 325816 p T + 53078032928 p^{2} T^{2} +$$$$85\!\cdots\!72$$$$T^{3} +$$$$63\!\cdots\!79$$$$T^{4} +$$$$35\!\cdots\!16$$$$T^{5} +$$$$17\!\cdots\!16$$$$T^{6} +$$$$35\!\cdots\!16$$$$p^{8} T^{7} +$$$$63\!\cdots\!79$$$$p^{16} T^{8} +$$$$85\!\cdots\!72$$$$p^{24} T^{9} + 53078032928 p^{34} T^{10} + 325816 p^{41} T^{11} + p^{48} T^{12}$$
53 $$1 + 13509122 T + 91248188605442 T^{2} +$$$$98\!\cdots\!42$$$$T^{3} +$$$$11\!\cdots\!79$$$$T^{4} +$$$$81\!\cdots\!76$$$$T^{5} +$$$$50\!\cdots\!36$$$$T^{6} +$$$$81\!\cdots\!76$$$$p^{8} T^{7} +$$$$11\!\cdots\!79$$$$p^{16} T^{8} +$$$$98\!\cdots\!42$$$$p^{24} T^{9} + 91248188605442 p^{32} T^{10} + 13509122 p^{40} T^{11} + p^{48} T^{12}$$
59 $$1 - 413223229068726 T^{2} +$$$$98\!\cdots\!15$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$98\!\cdots\!15$$$$p^{16} T^{8} - 413223229068726 p^{32} T^{10} + p^{48} T^{12}$$
61 $$( 1 - 12055596 T + 200152007609415 T^{2} +$$$$24\!\cdots\!80$$$$T^{3} + 200152007609415 p^{8} T^{4} - 12055596 p^{16} T^{5} + p^{24} T^{6} )^{2}$$
67 $$1 + 32827752 T + 538830650686752 T^{2} +$$$$16\!\cdots\!32$$$$T^{3} +$$$$54\!\cdots\!19$$$$T^{4} +$$$$88\!\cdots\!76$$$$T^{5} +$$$$12\!\cdots\!76$$$$T^{6} +$$$$88\!\cdots\!76$$$$p^{8} T^{7} +$$$$54\!\cdots\!19$$$$p^{16} T^{8} +$$$$16\!\cdots\!32$$$$p^{24} T^{9} + 538830650686752 p^{32} T^{10} + 32827752 p^{40} T^{11} + p^{48} T^{12}$$
71 $$( 1 + 6996464 T + 1071469029384215 T^{2} +$$$$14\!\cdots\!80$$$$T^{3} + 1071469029384215 p^{8} T^{4} + 6996464 p^{16} T^{5} + p^{24} T^{6} )^{2}$$
73 $$1 - 111859638 T + 6256289306745522 T^{2} -$$$$29\!\cdots\!78$$$$T^{3} +$$$$12\!\cdots\!39$$$$T^{4} -$$$$42\!\cdots\!64$$$$T^{5} +$$$$12\!\cdots\!16$$$$T^{6} -$$$$42\!\cdots\!64$$$$p^{8} T^{7} +$$$$12\!\cdots\!39$$$$p^{16} T^{8} -$$$$29\!\cdots\!78$$$$p^{24} T^{9} + 6256289306745522 p^{32} T^{10} - 111859638 p^{40} T^{11} + p^{48} T^{12}$$
79 $$1 - 4185433961698566 T^{2} +$$$$55\!\cdots\!15$$$$T^{4} -$$$$46\!\cdots\!20$$$$T^{6} +$$$$55\!\cdots\!15$$$$p^{16} T^{8} - 4185433961698566 p^{32} T^{10} + p^{48} T^{12}$$
83 $$1 + 14768432 T + 109053291869312 T^{2} +$$$$28\!\cdots\!12$$$$T^{3} +$$$$10\!\cdots\!19$$$$T^{4} +$$$$10\!\cdots\!16$$$$T^{5} +$$$$83\!\cdots\!56$$$$T^{6} +$$$$10\!\cdots\!16$$$$p^{8} T^{7} +$$$$10\!\cdots\!19$$$$p^{16} T^{8} +$$$$28\!\cdots\!12$$$$p^{24} T^{9} + 109053291869312 p^{32} T^{10} + 14768432 p^{40} T^{11} + p^{48} T^{12}$$
89 $$1 - 8165894455313286 T^{2} +$$$$62\!\cdots\!15$$$$T^{4} -$$$$26\!\cdots\!20$$$$T^{6} +$$$$62\!\cdots\!15$$$$p^{16} T^{8} - 8165894455313286 p^{32} T^{10} + p^{48} T^{12}$$
97 $$1 + 186656202 T + 17420268872532402 T^{2} +$$$$93\!\cdots\!22$$$$T^{3} +$$$$66\!\cdots\!79$$$$T^{4} +$$$$11\!\cdots\!16$$$$T^{5} +$$$$13\!\cdots\!16$$$$T^{6} +$$$$11\!\cdots\!16$$$$p^{8} T^{7} +$$$$66\!\cdots\!79$$$$p^{16} T^{8} +$$$$93\!\cdots\!22$$$$p^{24} T^{9} + 17420268872532402 p^{32} T^{10} + 186656202 p^{40} T^{11} + p^{48} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$