# Properties

 Label 8-45e4-1.1-c21e4-0-0 Degree $8$ Conductor $4100625$ Sign $1$ Analytic cond. $2.50170\times 10^{8}$ Root an. cond. $11.2144$ Motivic weight $21$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 897·2-s − 4.37e6·4-s − 3.90e7·5-s − 2.34e8·7-s − 4.81e9·8-s − 3.50e10·10-s − 3.14e10·11-s − 2.70e10·13-s − 2.10e11·14-s + 6.70e12·16-s + 2.94e12·17-s − 2.42e13·19-s + 1.70e14·20-s − 2.82e13·22-s − 1.03e13·23-s + 9.53e14·25-s − 2.42e13·26-s + 1.02e15·28-s − 4.72e15·29-s − 1.09e15·31-s + 8.96e15·32-s + 2.64e15·34-s + 9.16e15·35-s + 4.81e15·37-s − 2.17e16·38-s + 1.88e17·40-s − 2.89e17·41-s + ⋯
 L(s)  = 1 + 0.619·2-s − 2.08·4-s − 1.78·5-s − 0.313·7-s − 1.58·8-s − 1.10·10-s − 0.366·11-s − 0.0543·13-s − 0.194·14-s + 1.52·16-s + 0.354·17-s − 0.908·19-s + 3.73·20-s − 0.226·22-s − 0.0520·23-s + 2·25-s − 0.0336·26-s + 0.654·28-s − 2.08·29-s − 0.239·31-s + 1.40·32-s + 0.219·34-s + 0.561·35-s + 0.164·37-s − 0.562·38-s + 2.83·40-s − 3.37·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$4100625$$    =    $$3^{8} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$2.50170\times 10^{8}$$ Root analytic conductor: $$11.2144$$ Motivic weight: $$21$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 4100625,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)$$

## Particular Values

 $$L(11)$$ $$\approx$$ $$0.01920876667$$ $$L(\frac12)$$ $$\approx$$ $$0.01920876667$$ $$L(\frac{23}{2})$$ 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)$
bad3 $$1$$
5$C_1$ $$( 1 + p^{10} T )^{4}$$
good2$C_2 \wr S_4$ $$1 - 897 T + 2589085 p T^{2} - 117236669 p^{5} T^{3} + 14637391839 p^{10} T^{4} - 117236669 p^{26} T^{5} + 2589085 p^{43} T^{6} - 897 p^{63} T^{7} + p^{84} T^{8}$$
7$C_2 \wr S_4$ $$1 + 4787296 p^{2} T + 7592135803176028 p^{2} T^{2} -$$$$13\!\cdots\!36$$$$p^{4} T^{3} +$$$$73\!\cdots\!10$$$$p^{4} T^{4} -$$$$13\!\cdots\!36$$$$p^{25} T^{5} + 7592135803176028 p^{44} T^{6} + 4787296 p^{65} T^{7} + p^{84} T^{8}$$
11$C_2 \wr S_4$ $$1 + 31491830256 T +$$$$93\!\cdots\!92$$$$p T^{2} +$$$$46\!\cdots\!00$$$$p^{2} T^{3} +$$$$53\!\cdots\!06$$$$p^{3} T^{4} +$$$$46\!\cdots\!00$$$$p^{23} T^{5} +$$$$93\!\cdots\!92$$$$p^{43} T^{6} + 31491830256 p^{63} T^{7} + p^{84} T^{8}$$
13$C_2 \wr S_4$ $$1 + 27017977768 T -$$$$92\!\cdots\!72$$$$p T^{2} +$$$$33\!\cdots\!24$$$$p^{2} T^{3} +$$$$39\!\cdots\!30$$$$p^{3} T^{4} +$$$$33\!\cdots\!24$$$$p^{23} T^{5} -$$$$92\!\cdots\!72$$$$p^{43} T^{6} + 27017977768 p^{63} T^{7} + p^{84} T^{8}$$
17$C_2 \wr S_4$ $$1 - 2946095028888 T +$$$$67\!\cdots\!16$$$$p T^{2} -$$$$25\!\cdots\!20$$$$p^{2} T^{3} +$$$$19\!\cdots\!02$$$$p^{3} T^{4} -$$$$25\!\cdots\!20$$$$p^{23} T^{5} +$$$$67\!\cdots\!16$$$$p^{43} T^{6} - 2946095028888 p^{63} T^{7} + p^{84} T^{8}$$
19$C_2 \wr S_4$ $$1 + 24270353300752 T +$$$$48\!\cdots\!92$$$$p^{2} T^{2} +$$$$36\!\cdots\!36$$$$p^{3} T^{3} +$$$$21\!\cdots\!26$$$$p^{3} T^{4} +$$$$36\!\cdots\!36$$$$p^{24} T^{5} +$$$$48\!\cdots\!92$$$$p^{44} T^{6} + 24270353300752 p^{63} T^{7} + p^{84} T^{8}$$
23$C_2 \wr S_4$ $$1 + 10350924920928 T +$$$$70\!\cdots\!08$$$$T^{2} +$$$$70\!\cdots\!12$$$$T^{3} +$$$$23\!\cdots\!10$$$$T^{4} +$$$$70\!\cdots\!12$$$$p^{21} T^{5} +$$$$70\!\cdots\!08$$$$p^{42} T^{6} + 10350924920928 p^{63} T^{7} + p^{84} T^{8}$$
29$C_2 \wr S_4$ $$1 + 4728924677079096 T +$$$$11\!\cdots\!00$$$$T^{2} +$$$$17\!\cdots\!92$$$$T^{3} +$$$$17\!\cdots\!18$$$$T^{4} +$$$$17\!\cdots\!92$$$$p^{21} T^{5} +$$$$11\!\cdots\!00$$$$p^{42} T^{6} + 4728924677079096 p^{63} T^{7} + p^{84} T^{8}$$
31$C_2 \wr S_4$ $$1 + 1094923910405536 T +$$$$68\!\cdots\!48$$$$T^{2} +$$$$44\!\cdots\!48$$$$T^{3} +$$$$19\!\cdots\!54$$$$T^{4} +$$$$44\!\cdots\!48$$$$p^{21} T^{5} +$$$$68\!\cdots\!48$$$$p^{42} T^{6} + 1094923910405536 p^{63} T^{7} + p^{84} T^{8}$$
37$C_2 \wr S_4$ $$1 - 4813435696247096 T +$$$$17\!\cdots\!12$$$$T^{2} -$$$$33\!\cdots\!36$$$$T^{3} +$$$$15\!\cdots\!50$$$$T^{4} -$$$$33\!\cdots\!36$$$$p^{21} T^{5} +$$$$17\!\cdots\!12$$$$p^{42} T^{6} - 4813435696247096 p^{63} T^{7} + p^{84} T^{8}$$
41$C_2 \wr S_4$ $$1 + 289731591445930344 T +$$$$49\!\cdots\!68$$$$T^{2} +$$$$57\!\cdots\!52$$$$T^{3} +$$$$55\!\cdots\!14$$$$T^{4} +$$$$57\!\cdots\!52$$$$p^{21} T^{5} +$$$$49\!\cdots\!68$$$$p^{42} T^{6} + 289731591445930344 p^{63} T^{7} + p^{84} T^{8}$$
43$C_2 \wr S_4$ $$1 - 451091912658458000 T +$$$$13\!\cdots\!00$$$$T^{2} -$$$$27\!\cdots\!00$$$$T^{3} +$$$$44\!\cdots\!98$$$$T^{4} -$$$$27\!\cdots\!00$$$$p^{21} T^{5} +$$$$13\!\cdots\!00$$$$p^{42} T^{6} - 451091912658458000 p^{63} T^{7} + p^{84} T^{8}$$
47$C_2 \wr S_4$ $$1 + 813883435638492480 T +$$$$59\!\cdots\!20$$$$T^{2} +$$$$27\!\cdots\!60$$$$T^{3} +$$$$11\!\cdots\!18$$$$T^{4} +$$$$27\!\cdots\!60$$$$p^{21} T^{5} +$$$$59\!\cdots\!20$$$$p^{42} T^{6} + 813883435638492480 p^{63} T^{7} + p^{84} T^{8}$$
53$C_2 \wr S_4$ $$1 + 697278335404085208 T +$$$$47\!\cdots\!28$$$$T^{2} +$$$$22\!\cdots\!72$$$$T^{3} +$$$$10\!\cdots\!50$$$$T^{4} +$$$$22\!\cdots\!72$$$$p^{21} T^{5} +$$$$47\!\cdots\!28$$$$p^{42} T^{6} + 697278335404085208 p^{63} T^{7} + p^{84} T^{8}$$
59$C_2 \wr S_4$ $$1 + 6622888614569598192 T +$$$$40\!\cdots\!72$$$$T^{2} +$$$$20\!\cdots\!04$$$$T^{3} +$$$$88\!\cdots\!34$$$$T^{4} +$$$$20\!\cdots\!04$$$$p^{21} T^{5} +$$$$40\!\cdots\!72$$$$p^{42} T^{6} + 6622888614569598192 p^{63} T^{7} + p^{84} T^{8}$$
61$C_2 \wr S_4$ $$1 - 7390887218011683320 T +$$$$30\!\cdots\!96$$$$T^{2} -$$$$27\!\cdots\!80$$$$T^{3} +$$$$27\!\cdots\!46$$$$T^{4} -$$$$27\!\cdots\!80$$$$p^{21} T^{5} +$$$$30\!\cdots\!96$$$$p^{42} T^{6} - 7390887218011683320 p^{63} T^{7} + p^{84} T^{8}$$
67$C_2 \wr S_4$ $$1 + 24188188449376788688 T +$$$$66\!\cdots\!72$$$$T^{2} +$$$$12\!\cdots\!80$$$$T^{3} +$$$$22\!\cdots\!26$$$$T^{4} +$$$$12\!\cdots\!80$$$$p^{21} T^{5} +$$$$66\!\cdots\!72$$$$p^{42} T^{6} + 24188188449376788688 p^{63} T^{7} + p^{84} T^{8}$$
71$C_2 \wr S_4$ $$1 - 37390337803999713312 T +$$$$18\!\cdots\!88$$$$T^{2} -$$$$49\!\cdots\!64$$$$T^{3} +$$$$15\!\cdots\!70$$$$T^{4} -$$$$49\!\cdots\!64$$$$p^{21} T^{5} +$$$$18\!\cdots\!88$$$$p^{42} T^{6} - 37390337803999713312 p^{63} T^{7} + p^{84} T^{8}$$
73$C_2 \wr S_4$ $$1 + 37253672904265201432 T +$$$$53\!\cdots\!28$$$$T^{2} +$$$$13\!\cdots\!88$$$$T^{3} +$$$$10\!\cdots\!30$$$$T^{4} +$$$$13\!\cdots\!88$$$$p^{21} T^{5} +$$$$53\!\cdots\!28$$$$p^{42} T^{6} + 37253672904265201432 p^{63} T^{7} + p^{84} T^{8}$$
79$C_2 \wr S_4$ $$1 +$$$$10\!\cdots\!00$$$$T +$$$$20\!\cdots\!16$$$$T^{2} +$$$$13\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!46$$$$T^{4} +$$$$13\!\cdots\!00$$$$p^{21} T^{5} +$$$$20\!\cdots\!16$$$$p^{42} T^{6} +$$$$10\!\cdots\!00$$$$p^{63} T^{7} + p^{84} T^{8}$$
83$C_2 \wr S_4$ $$1 -$$$$14\!\cdots\!96$$$$T +$$$$49\!\cdots\!60$$$$T^{2} -$$$$54\!\cdots\!56$$$$T^{3} +$$$$11\!\cdots\!26$$$$T^{4} -$$$$54\!\cdots\!56$$$$p^{21} T^{5} +$$$$49\!\cdots\!60$$$$p^{42} T^{6} -$$$$14\!\cdots\!96$$$$p^{63} T^{7} + p^{84} T^{8}$$
89$C_2 \wr S_4$ $$1 +$$$$55\!\cdots\!48$$$$T +$$$$43\!\cdots\!52$$$$T^{2} +$$$$14\!\cdots\!96$$$$T^{3} +$$$$60\!\cdots\!14$$$$T^{4} +$$$$14\!\cdots\!96$$$$p^{21} T^{5} +$$$$43\!\cdots\!52$$$$p^{42} T^{6} +$$$$55\!\cdots\!48$$$$p^{63} T^{7} + p^{84} T^{8}$$
97$C_2 \wr S_4$ $$1 +$$$$23\!\cdots\!28$$$$T +$$$$13\!\cdots\!32$$$$T^{2} +$$$$61\!\cdots\!20$$$$T^{3} +$$$$80\!\cdots\!66$$$$T^{4} +$$$$61\!\cdots\!20$$$$p^{21} T^{5} +$$$$13\!\cdots\!32$$$$p^{42} T^{6} +$$$$23\!\cdots\!28$$$$p^{63} T^{7} + p^{84} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$