Properties

 Degree 10 Conductor $5^{5}$ Sign $-1$ Motivic weight 33 Primitive no Self-dual yes Analytic rank 5

Origins of factors

Dirichlet series

 L(s)  = 1 + 3.04e4·2-s − 1.49e7·3-s − 2.04e10·4-s − 7.62e11·5-s − 4.56e11·6-s − 6.54e13·7-s − 6.67e14·8-s − 6.51e15·9-s − 2.32e16·10-s − 2.87e17·11-s + 3.06e17·12-s + 2.39e18·13-s − 1.99e18·14-s + 1.14e19·15-s + 1.82e20·16-s − 3.08e20·17-s − 1.98e20·18-s + 9.18e20·19-s + 1.55e22·20-s + 9.81e20·21-s − 8.76e21·22-s + 5.54e22·23-s + 1.00e22·24-s + 3.49e23·25-s + 7.30e22·26-s + 3.52e23·27-s + 1.33e24·28-s + ⋯
 L(s)  = 1 + 0.328·2-s − 0.201·3-s − 2.37·4-s − 2.23·5-s − 0.0660·6-s − 0.744·7-s − 0.838·8-s − 1.17·9-s − 0.735·10-s − 1.88·11-s + 0.478·12-s + 0.999·13-s − 0.244·14-s + 0.449·15-s + 2.48·16-s − 1.53·17-s − 0.385·18-s + 0.730·19-s + 5.32·20-s + 0.149·21-s − 0.620·22-s + 1.88·23-s + 0.168·24-s + 3·25-s + 0.328·26-s + 0.851·27-s + 1.77·28-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s+33/2)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$10$$ $$N$$ = $$3125$$    =    $$5^{5}$$ $$\varepsilon$$ = $-1$ motivic weight = $$33$$ character : induced by $\chi_{5} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 5 Selberg data = $(10,\ 3125,\ (\ :33/2, 33/2, 33/2, 33/2, 33/2),\ -1)$ $L(17)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{35}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 10. If $p = 5$, then $F_p$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$ $$( 1 + p^{16} T )^{5}$$
good2$C_2 \wr S_5$ $$1 - 3809 p^{3} T + 667764131 p^{5} T^{2} - 148029135825 p^{12} T^{3} + 480463746951481 p^{19} T^{4} - 23561651058556561 p^{28} T^{5} + 480463746951481 p^{52} T^{6} - 148029135825 p^{78} T^{7} + 667764131 p^{104} T^{8} - 3809 p^{135} T^{9} + p^{165} T^{10}$$
3$C_2 \wr S_5$ $$1 + 4996238 p T + 83201934504583 p^{4} T^{2} - 7840773217846951400 p^{9} T^{3} +$$$$13\!\cdots\!94$$$$p^{15} T^{4} -$$$$54\!\cdots\!32$$$$p^{22} T^{5} +$$$$13\!\cdots\!94$$$$p^{48} T^{6} - 7840773217846951400 p^{75} T^{7} + 83201934504583 p^{103} T^{8} + 4996238 p^{133} T^{9} + p^{165} T^{10}$$
7$C_2 \wr S_5$ $$1 + 9350365969594 p T +$$$$11\!\cdots\!01$$$$p^{5} T^{2} +$$$$50\!\cdots\!00$$$$p^{3} T^{3} +$$$$22\!\cdots\!02$$$$p^{6} T^{4} +$$$$56\!\cdots\!16$$$$p^{10} T^{5} +$$$$22\!\cdots\!02$$$$p^{39} T^{6} +$$$$50\!\cdots\!00$$$$p^{69} T^{7} +$$$$11\!\cdots\!01$$$$p^{104} T^{8} + 9350365969594 p^{133} T^{9} + p^{165} T^{10}$$
11$C_2 \wr S_5$ $$1 + 287801801877976640 T +$$$$10\!\cdots\!45$$$$p T^{2} +$$$$19\!\cdots\!80$$$$p^{2} T^{3} +$$$$40\!\cdots\!10$$$$p^{3} T^{4} +$$$$53\!\cdots\!28$$$$p^{4} T^{5} +$$$$40\!\cdots\!10$$$$p^{36} T^{6} +$$$$19\!\cdots\!80$$$$p^{68} T^{7} +$$$$10\!\cdots\!45$$$$p^{100} T^{8} + 287801801877976640 p^{132} T^{9} + p^{165} T^{10}$$
13$C_2 \wr S_5$ $$1 - 184400088529992882 p T +$$$$15\!\cdots\!81$$$$p T^{2} -$$$$25\!\cdots\!00$$$$p^{2} T^{3} +$$$$85\!\cdots\!94$$$$p^{3} T^{4} -$$$$96\!\cdots\!16$$$$p^{5} T^{5} +$$$$85\!\cdots\!94$$$$p^{36} T^{6} -$$$$25\!\cdots\!00$$$$p^{68} T^{7} +$$$$15\!\cdots\!81$$$$p^{100} T^{8} - 184400088529992882 p^{133} T^{9} + p^{165} T^{10}$$
17$C_2 \wr S_5$ $$1 +$$$$30\!\cdots\!18$$$$T +$$$$15\!\cdots\!37$$$$T^{2} +$$$$31\!\cdots\!00$$$$T^{3} +$$$$62\!\cdots\!14$$$$p T^{4} +$$$$61\!\cdots\!56$$$$p^{2} T^{5} +$$$$62\!\cdots\!14$$$$p^{34} T^{6} +$$$$31\!\cdots\!00$$$$p^{66} T^{7} +$$$$15\!\cdots\!37$$$$p^{99} T^{8} +$$$$30\!\cdots\!18$$$$p^{132} T^{9} + p^{165} T^{10}$$
19$C_2 \wr S_5$ $$1 -$$$$91\!\cdots\!00$$$$T +$$$$53\!\cdots\!95$$$$T^{2} -$$$$37\!\cdots\!00$$$$T^{3} +$$$$74\!\cdots\!90$$$$p T^{4} -$$$$11\!\cdots\!00$$$$p^{3} T^{5} +$$$$74\!\cdots\!90$$$$p^{34} T^{6} -$$$$37\!\cdots\!00$$$$p^{66} T^{7} +$$$$53\!\cdots\!95$$$$p^{99} T^{8} -$$$$91\!\cdots\!00$$$$p^{132} T^{9} + p^{165} T^{10}$$
23$C_2 \wr S_5$ $$1 -$$$$55\!\cdots\!46$$$$T +$$$$73\!\cdots\!21$$$$p T^{2} -$$$$10\!\cdots\!00$$$$p^{2} T^{3} +$$$$88\!\cdots\!58$$$$p^{4} T^{4} -$$$$30\!\cdots\!68$$$$p^{4} T^{5} +$$$$88\!\cdots\!58$$$$p^{37} T^{6} -$$$$10\!\cdots\!00$$$$p^{68} T^{7} +$$$$73\!\cdots\!21$$$$p^{100} T^{8} -$$$$55\!\cdots\!46$$$$p^{132} T^{9} + p^{165} T^{10}$$
29$C_2 \wr S_5$ $$1 +$$$$31\!\cdots\!50$$$$p T +$$$$99\!\cdots\!45$$$$p^{2} T^{2} +$$$$24\!\cdots\!00$$$$p^{3} T^{3} +$$$$40\!\cdots\!10$$$$p^{4} T^{4} +$$$$75\!\cdots\!00$$$$p^{5} T^{5} +$$$$40\!\cdots\!10$$$$p^{37} T^{6} +$$$$24\!\cdots\!00$$$$p^{69} T^{7} +$$$$99\!\cdots\!45$$$$p^{101} T^{8} +$$$$31\!\cdots\!50$$$$p^{133} T^{9} + p^{165} T^{10}$$
31$C_2 \wr S_5$ $$1 +$$$$13\!\cdots\!40$$$$T +$$$$13\!\cdots\!45$$$$p T^{2} -$$$$17\!\cdots\!20$$$$p^{2} T^{3} +$$$$23\!\cdots\!10$$$$p^{3} T^{4} -$$$$13\!\cdots\!12$$$$p^{4} T^{5} +$$$$23\!\cdots\!10$$$$p^{36} T^{6} -$$$$17\!\cdots\!20$$$$p^{68} T^{7} +$$$$13\!\cdots\!45$$$$p^{100} T^{8} +$$$$13\!\cdots\!40$$$$p^{132} T^{9} + p^{165} T^{10}$$
37$C_2 \wr S_5$ $$1 +$$$$14\!\cdots\!38$$$$T +$$$$33\!\cdots\!97$$$$T^{2} +$$$$32\!\cdots\!00$$$$T^{3} +$$$$40\!\cdots\!18$$$$T^{4} +$$$$27\!\cdots\!84$$$$T^{5} +$$$$40\!\cdots\!18$$$$p^{33} T^{6} +$$$$32\!\cdots\!00$$$$p^{66} T^{7} +$$$$33\!\cdots\!97$$$$p^{99} T^{8} +$$$$14\!\cdots\!38$$$$p^{132} T^{9} + p^{165} T^{10}$$
41$C_2 \wr S_5$ $$1 +$$$$13\!\cdots\!90$$$$T +$$$$13\!\cdots\!45$$$$T^{2} +$$$$98\!\cdots\!80$$$$T^{3} +$$$$56\!\cdots\!10$$$$T^{4} +$$$$25\!\cdots\!48$$$$T^{5} +$$$$56\!\cdots\!10$$$$p^{33} T^{6} +$$$$98\!\cdots\!80$$$$p^{66} T^{7} +$$$$13\!\cdots\!45$$$$p^{99} T^{8} +$$$$13\!\cdots\!90$$$$p^{132} T^{9} + p^{165} T^{10}$$
43$C_2 \wr S_5$ $$1 +$$$$23\!\cdots\!94$$$$T +$$$$38\!\cdots\!43$$$$T^{2} +$$$$49\!\cdots\!00$$$$T^{3} +$$$$56\!\cdots\!98$$$$T^{4} +$$$$55\!\cdots\!12$$$$T^{5} +$$$$56\!\cdots\!98$$$$p^{33} T^{6} +$$$$49\!\cdots\!00$$$$p^{66} T^{7} +$$$$38\!\cdots\!43$$$$p^{99} T^{8} +$$$$23\!\cdots\!94$$$$p^{132} T^{9} + p^{165} T^{10}$$
47$C_2 \wr S_5$ $$1 +$$$$77\!\cdots\!98$$$$T +$$$$80\!\cdots\!27$$$$T^{2} +$$$$42\!\cdots\!00$$$$T^{3} +$$$$25\!\cdots\!58$$$$T^{4} +$$$$94\!\cdots\!84$$$$T^{5} +$$$$25\!\cdots\!58$$$$p^{33} T^{6} +$$$$42\!\cdots\!00$$$$p^{66} T^{7} +$$$$80\!\cdots\!27$$$$p^{99} T^{8} +$$$$77\!\cdots\!98$$$$p^{132} T^{9} + p^{165} T^{10}$$
53$C_2 \wr S_5$ $$1 -$$$$10\!\cdots\!86$$$$T +$$$$65\!\cdots\!73$$$$T^{2} -$$$$29\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!58$$$$T^{4} -$$$$33\!\cdots\!88$$$$T^{5} +$$$$10\!\cdots\!58$$$$p^{33} T^{6} -$$$$29\!\cdots\!00$$$$p^{66} T^{7} +$$$$65\!\cdots\!73$$$$p^{99} T^{8} -$$$$10\!\cdots\!86$$$$p^{132} T^{9} + p^{165} T^{10}$$
59$C_2 \wr S_5$ $$1 -$$$$25\!\cdots\!00$$$$T +$$$$12\!\cdots\!95$$$$T^{2} -$$$$24\!\cdots\!00$$$$T^{3} +$$$$62\!\cdots\!10$$$$T^{4} -$$$$93\!\cdots\!00$$$$T^{5} +$$$$62\!\cdots\!10$$$$p^{33} T^{6} -$$$$24\!\cdots\!00$$$$p^{66} T^{7} +$$$$12\!\cdots\!95$$$$p^{99} T^{8} -$$$$25\!\cdots\!00$$$$p^{132} T^{9} + p^{165} T^{10}$$
61$C_2 \wr S_5$ $$1 -$$$$74\!\cdots\!10$$$$T +$$$$40\!\cdots\!45$$$$T^{2} -$$$$17\!\cdots\!20$$$$T^{3} +$$$$62\!\cdots\!10$$$$T^{4} -$$$$18\!\cdots\!52$$$$T^{5} +$$$$62\!\cdots\!10$$$$p^{33} T^{6} -$$$$17\!\cdots\!20$$$$p^{66} T^{7} +$$$$40\!\cdots\!45$$$$p^{99} T^{8} -$$$$74\!\cdots\!10$$$$p^{132} T^{9} + p^{165} T^{10}$$
67$C_2 \wr S_5$ $$1 +$$$$37\!\cdots\!18$$$$T +$$$$13\!\cdots\!87$$$$T^{2} +$$$$29\!\cdots\!00$$$$T^{3} +$$$$58\!\cdots\!38$$$$T^{4} +$$$$81\!\cdots\!84$$$$T^{5} +$$$$58\!\cdots\!38$$$$p^{33} T^{6} +$$$$29\!\cdots\!00$$$$p^{66} T^{7} +$$$$13\!\cdots\!87$$$$p^{99} T^{8} +$$$$37\!\cdots\!18$$$$p^{132} T^{9} + p^{165} T^{10}$$
71$C_2 \wr S_5$ $$1 +$$$$55\!\cdots\!40$$$$T +$$$$31\!\cdots\!95$$$$T^{2} -$$$$69\!\cdots\!20$$$$T^{3} -$$$$24\!\cdots\!90$$$$T^{4} -$$$$22\!\cdots\!52$$$$T^{5} -$$$$24\!\cdots\!90$$$$p^{33} T^{6} -$$$$69\!\cdots\!20$$$$p^{66} T^{7} +$$$$31\!\cdots\!95$$$$p^{99} T^{8} +$$$$55\!\cdots\!40$$$$p^{132} T^{9} + p^{165} T^{10}$$
73$C_2 \wr S_5$ $$1 +$$$$13\!\cdots\!54$$$$T +$$$$13\!\cdots\!33$$$$T^{2} +$$$$73\!\cdots\!00$$$$T^{3} +$$$$35\!\cdots\!78$$$$T^{4} +$$$$14\!\cdots\!12$$$$T^{5} +$$$$35\!\cdots\!78$$$$p^{33} T^{6} +$$$$73\!\cdots\!00$$$$p^{66} T^{7} +$$$$13\!\cdots\!33$$$$p^{99} T^{8} +$$$$13\!\cdots\!54$$$$p^{132} T^{9} + p^{165} T^{10}$$
79$C_2 \wr S_5$ $$1 +$$$$10\!\cdots\!00$$$$T +$$$$74\!\cdots\!95$$$$T^{2} -$$$$86\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!10$$$$T^{4} -$$$$82\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!10$$$$p^{33} T^{6} -$$$$86\!\cdots\!00$$$$p^{66} T^{7} +$$$$74\!\cdots\!95$$$$p^{99} T^{8} +$$$$10\!\cdots\!00$$$$p^{132} T^{9} + p^{165} T^{10}$$
83$C_2 \wr S_5$ $$1 -$$$$92\!\cdots\!26$$$$T +$$$$11\!\cdots\!63$$$$T^{2} -$$$$75\!\cdots\!00$$$$T^{3} +$$$$51\!\cdots\!38$$$$T^{4} -$$$$23\!\cdots\!88$$$$T^{5} +$$$$51\!\cdots\!38$$$$p^{33} T^{6} -$$$$75\!\cdots\!00$$$$p^{66} T^{7} +$$$$11\!\cdots\!63$$$$p^{99} T^{8} -$$$$92\!\cdots\!26$$$$p^{132} T^{9} + p^{165} T^{10}$$
89$C_2 \wr S_5$ $$1 +$$$$60\!\cdots\!50$$$$T +$$$$56\!\cdots\!45$$$$T^{2} +$$$$56\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!10$$$$T^{4} +$$$$16\!\cdots\!00$$$$T^{5} +$$$$18\!\cdots\!10$$$$p^{33} T^{6} +$$$$56\!\cdots\!00$$$$p^{66} T^{7} +$$$$56\!\cdots\!45$$$$p^{99} T^{8} +$$$$60\!\cdots\!50$$$$p^{132} T^{9} + p^{165} T^{10}$$
97$C_2 \wr S_5$ $$1 +$$$$86\!\cdots\!98$$$$T +$$$$15\!\cdots\!77$$$$T^{2} +$$$$11\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!58$$$$T^{4} +$$$$62\!\cdots\!84$$$$T^{5} +$$$$10\!\cdots\!58$$$$p^{33} T^{6} +$$$$11\!\cdots\!00$$$$p^{66} T^{7} +$$$$15\!\cdots\!77$$$$p^{99} T^{8} +$$$$86\!\cdots\!98$$$$p^{132} T^{9} + p^{165} T^{10}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}