# Properties

 Label 10-280e5-1.1-c5e5-0-1 Degree $10$ Conductor $1.721\times 10^{12}$ Sign $1$ Analytic cond. $1.82638\times 10^{8}$ Root an. cond. $6.70130$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 15·3-s + 125·5-s + 245·7-s + 71·9-s + 281·11-s + 909·13-s + 1.87e3·15-s + 1.49e3·17-s − 422·19-s + 3.67e3·21-s − 62·23-s + 9.37e3·25-s − 2.39e3·27-s − 2.04e3·29-s + 1.63e3·31-s + 4.21e3·33-s + 3.06e4·35-s − 1.03e4·37-s + 1.36e4·39-s + 6.42e3·41-s + 2.83e4·43-s + 8.87e3·45-s + 2.09e4·47-s + 3.60e4·49-s + 2.24e4·51-s + 4.37e4·53-s + 3.51e4·55-s + ⋯
 L(s)  = 1 + 0.962·3-s + 2.23·5-s + 1.88·7-s + 0.292·9-s + 0.700·11-s + 1.49·13-s + 2.15·15-s + 1.25·17-s − 0.268·19-s + 1.81·21-s − 0.0244·23-s + 3·25-s − 0.632·27-s − 0.451·29-s + 0.305·31-s + 0.673·33-s + 4.22·35-s − 1.24·37-s + 1.43·39-s + 0.596·41-s + 2.33·43-s + 0.653·45-s + 1.38·47-s + 15/7·49-s + 1.20·51-s + 2.13·53-s + 1.56·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{5} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{5} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$10$$ Conductor: $$2^{15} \cdot 5^{5} \cdot 7^{5}$$ Sign: $1$ Analytic conductor: $$1.82638\times 10^{8}$$ Root analytic conductor: $$6.70130$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{280} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(10,\ 2^{15} \cdot 5^{5} \cdot 7^{5} ,\ ( \ : 5/2, 5/2, 5/2, 5/2, 5/2 ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$82.78089068$$ $$L(\frac12)$$ $$\approx$$ $$82.78089068$$ $$L(\frac{7}{2})$$ 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)$
bad2 $$1$$
5$C_1$ $$( 1 - p^{2} T )^{5}$$
7$C_1$ $$( 1 - p^{2} T )^{5}$$
good3$C_2 \wr S_5$ $$1 - 5 p T + 154 T^{2} + 1151 T^{3} - 17969 p T^{4} + 63736 p^{2} T^{5} - 17969 p^{6} T^{6} + 1151 p^{10} T^{7} + 154 p^{15} T^{8} - 5 p^{21} T^{9} + p^{25} T^{10}$$
11$C_2 \wr S_5$ $$1 - 281 T + 221602 T^{2} + 4570105 T^{3} + 40747949317 T^{4} - 4335881998760 T^{5} + 40747949317 p^{5} T^{6} + 4570105 p^{10} T^{7} + 221602 p^{15} T^{8} - 281 p^{20} T^{9} + p^{25} T^{10}$$
13$C_2 \wr S_5$ $$1 - 909 T + 1436620 T^{2} - 858523871 T^{3} + 839117392375 T^{4} - 396610941659336 T^{5} + 839117392375 p^{5} T^{6} - 858523871 p^{10} T^{7} + 1436620 p^{15} T^{8} - 909 p^{20} T^{9} + p^{25} T^{10}$$
17$C_2 \wr S_5$ $$1 - 1495 T + 5368176 T^{2} - 6041848165 T^{3} + 13573892472707 T^{4} - 12001168625572880 T^{5} + 13573892472707 p^{5} T^{6} - 6041848165 p^{10} T^{7} + 5368176 p^{15} T^{8} - 1495 p^{20} T^{9} + p^{25} T^{10}$$
19$C_2 \wr S_5$ $$1 + 422 T + 390361 p T^{2} + 3236021232 T^{3} + 29599412304902 T^{4} + 12005015702197876 T^{5} + 29599412304902 p^{5} T^{6} + 3236021232 p^{10} T^{7} + 390361 p^{16} T^{8} + 422 p^{20} T^{9} + p^{25} T^{10}$$
23$C_2 \wr S_5$ $$1 + 62 T + 20580223 T^{2} - 2921201488 T^{3} + 222654385879558 T^{4} - 18821069520565660 T^{5} + 222654385879558 p^{5} T^{6} - 2921201488 p^{10} T^{7} + 20580223 p^{15} T^{8} + 62 p^{20} T^{9} + p^{25} T^{10}$$
29$C_2 \wr S_5$ $$1 + 2047 T + 12605620 T^{2} - 13927401731 T^{3} - 1908393559861 p T^{4} - 2737380069812364536 T^{5} - 1908393559861 p^{6} T^{6} - 13927401731 p^{10} T^{7} + 12605620 p^{15} T^{8} + 2047 p^{20} T^{9} + p^{25} T^{10}$$
31$C_2 \wr S_5$ $$1 - 1636 T + 109234699 T^{2} - 172511938992 T^{3} + 5563121129882810 T^{4} - 6956514622281569752 T^{5} + 5563121129882810 p^{5} T^{6} - 172511938992 p^{10} T^{7} + 109234699 p^{15} T^{8} - 1636 p^{20} T^{9} + p^{25} T^{10}$$
37$C_2 \wr S_5$ $$1 + 10358 T + 114652225 T^{2} + 201619793736 T^{3} + 4341551200446386 T^{4} + 446228899272382372 T^{5} + 4341551200446386 p^{5} T^{6} + 201619793736 p^{10} T^{7} + 114652225 p^{15} T^{8} + 10358 p^{20} T^{9} + p^{25} T^{10}$$
41$C_2 \wr S_5$ $$1 - 6424 T + 395117553 T^{2} - 646274781232 T^{3} + 60937741661540822 T^{4} + 34763395578700777680 T^{5} + 60937741661540822 p^{5} T^{6} - 646274781232 p^{10} T^{7} + 395117553 p^{15} T^{8} - 6424 p^{20} T^{9} + p^{25} T^{10}$$
43$C_2 \wr S_5$ $$1 - 28306 T + 642480547 T^{2} - 8887632516816 T^{3} + 119006154611192822 T^{4} -$$$$12\!\cdots\!16$$$$T^{5} + 119006154611192822 p^{5} T^{6} - 8887632516816 p^{10} T^{7} + 642480547 p^{15} T^{8} - 28306 p^{20} T^{9} + p^{25} T^{10}$$
47$C_2 \wr S_5$ $$1 - 20955 T + 1089343902 T^{2} - 18546466912749 T^{3} + 485237221486674017 T^{4} -$$$$63\!\cdots\!64$$$$T^{5} + 485237221486674017 p^{5} T^{6} - 18546466912749 p^{10} T^{7} + 1089343902 p^{15} T^{8} - 20955 p^{20} T^{9} + p^{25} T^{10}$$
53$C_2 \wr S_5$ $$1 - 43748 T + 838202445 T^{2} - 12691980106256 T^{3} + 562895427828965030 T^{4} -$$$$16\!\cdots\!60$$$$T^{5} + 562895427828965030 p^{5} T^{6} - 12691980106256 p^{10} T^{7} + 838202445 p^{15} T^{8} - 43748 p^{20} T^{9} + p^{25} T^{10}$$
59$C_2 \wr S_5$ $$1 - 45788 T + 3593077975 T^{2} - 121778928024912 T^{3} + 5136498549843134810 T^{4} -$$$$12\!\cdots\!44$$$$T^{5} + 5136498549843134810 p^{5} T^{6} - 121778928024912 p^{10} T^{7} + 3593077975 p^{15} T^{8} - 45788 p^{20} T^{9} + p^{25} T^{10}$$
61$C_2 \wr S_5$ $$1 - 50432 T + 4030546853 T^{2} - 136393865631120 T^{3} + 6194450556799903702 T^{4} -$$$$25\!\cdots\!36$$$$p T^{5} + 6194450556799903702 p^{5} T^{6} - 136393865631120 p^{10} T^{7} + 4030546853 p^{15} T^{8} - 50432 p^{20} T^{9} + p^{25} T^{10}$$
67$C_2 \wr S_5$ $$1 - 40712 T + 5343481519 T^{2} - 173526050513376 T^{3} + 12800459062427075018 T^{4} -$$$$32\!\cdots\!72$$$$T^{5} + 12800459062427075018 p^{5} T^{6} - 173526050513376 p^{10} T^{7} + 5343481519 p^{15} T^{8} - 40712 p^{20} T^{9} + p^{25} T^{10}$$
71$C_2 \wr S_5$ $$1 + 3096 T + 1680533635 T^{2} - 83824474204512 T^{3} + 6649208184641447530 T^{4} +$$$$25\!\cdots\!84$$$$T^{5} + 6649208184641447530 p^{5} T^{6} - 83824474204512 p^{10} T^{7} + 1680533635 p^{15} T^{8} + 3096 p^{20} T^{9} + p^{25} T^{10}$$
73$C_2 \wr S_5$ $$1 - 135438 T + 11904560997 T^{2} - 760960528321992 T^{3} + 42362862874340730882 T^{4} -$$$$20\!\cdots\!20$$$$T^{5} + 42362862874340730882 p^{5} T^{6} - 760960528321992 p^{10} T^{7} + 11904560997 p^{15} T^{8} - 135438 p^{20} T^{9} + p^{25} T^{10}$$
79$C_2 \wr S_5$ $$1 - 13191 T + 6068072158 T^{2} - 343248053940809 T^{3} + 16534271595589609465 T^{4} -$$$$17\!\cdots\!84$$$$T^{5} + 16534271595589609465 p^{5} T^{6} - 343248053940809 p^{10} T^{7} + 6068072158 p^{15} T^{8} - 13191 p^{20} T^{9} + p^{25} T^{10}$$
83$C_2 \wr S_5$ $$1 - 35108 T + 14240686639 T^{2} - 416149643153968 T^{3} + 97125793235490599674 T^{4} -$$$$23\!\cdots\!68$$$$T^{5} + 97125793235490599674 p^{5} T^{6} - 416149643153968 p^{10} T^{7} + 14240686639 p^{15} T^{8} - 35108 p^{20} T^{9} + p^{25} T^{10}$$
89$C_2 \wr S_5$ $$1 - 213772 T + 36142472577 T^{2} - 3766577124770512 T^{3} +$$$$36\!\cdots\!74$$$$T^{4} -$$$$26\!\cdots\!88$$$$T^{5} +$$$$36\!\cdots\!74$$$$p^{5} T^{6} - 3766577124770512 p^{10} T^{7} + 36142472577 p^{15} T^{8} - 213772 p^{20} T^{9} + p^{25} T^{10}$$
97$C_2 \wr S_5$ $$1 - 10659 T + 24686333912 T^{2} - 320233522422945 T^{3} +$$$$33\!\cdots\!39$$$$T^{4} -$$$$48\!\cdots\!36$$$$T^{5} +$$$$33\!\cdots\!39$$$$p^{5} T^{6} - 320233522422945 p^{10} T^{7} + 24686333912 p^{15} T^{8} - 10659 p^{20} T^{9} + p^{25} T^{10}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$