# Properties

 Label 8-55e4-1.1-c5e4-0-1 Degree $8$ Conductor $9150625$ Sign $1$ Analytic cond. $6054.70$ Root an. cond. $2.97003$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 5·2-s − 21·4-s − 100·5-s − 90·7-s + 155·8-s − 475·9-s + 500·10-s − 484·11-s + 820·13-s + 450·14-s − 1.02e3·16-s − 3.80e3·17-s + 2.37e3·18-s − 3.39e3·19-s + 2.10e3·20-s + 2.42e3·22-s − 3.02e3·23-s + 6.25e3·25-s − 4.10e3·26-s + 1.80e3·27-s + 1.89e3·28-s − 5.24e3·29-s + 4.73e3·31-s + 6.63e3·32-s + 1.90e4·34-s + 9.00e3·35-s + 9.97e3·36-s + ⋯
 L(s)  = 1 − 0.883·2-s − 0.656·4-s − 1.78·5-s − 0.694·7-s + 0.856·8-s − 1.95·9-s + 1.58·10-s − 1.20·11-s + 1.34·13-s + 0.613·14-s − 1.00·16-s − 3.18·17-s + 1.72·18-s − 2.15·19-s + 1.17·20-s + 1.06·22-s − 1.19·23-s + 2·25-s − 1.18·26-s + 0.475·27-s + 0.455·28-s − 1.15·29-s + 0.884·31-s + 1.14·32-s + 2.81·34-s + 1.24·35-s + 1.28·36-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$9150625$$    =    $$5^{4} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$6054.70$$ Root analytic conductor: $$2.97003$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 9150625,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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)$
bad5$C_1$ $$( 1 + p^{2} T )^{4}$$
11$C_1$ $$( 1 + p^{2} T )^{4}$$
good2$C_2 \wr S_4$ $$1 + 5 T + 23 p T^{2} + 45 p^{2} T^{3} + 265 p^{3} T^{4} + 45 p^{7} T^{5} + 23 p^{11} T^{6} + 5 p^{15} T^{7} + p^{20} T^{8}$$
3$C_2 \wr S_4$ $$1 + 475 T^{2} - 200 p^{2} T^{3} + 144856 T^{4} - 200 p^{7} T^{5} + 475 p^{10} T^{6} + p^{20} T^{8}$$
7$C_2 \wr S_4$ $$1 + 90 T + 3627 p T^{2} - 89470 p T^{3} + 229065672 T^{4} - 89470 p^{6} T^{5} + 3627 p^{11} T^{6} + 90 p^{15} T^{7} + p^{20} T^{8}$$
13$C_2 \wr S_4$ $$1 - 820 T + 1282272 T^{2} - 714676780 T^{3} + 671592391694 T^{4} - 714676780 p^{5} T^{5} + 1282272 p^{10} T^{6} - 820 p^{15} T^{7} + p^{20} T^{8}$$
17$C_2 \wr S_4$ $$1 + 3800 T + 9659155 T^{2} + 16711093200 T^{3} + 22897224473516 T^{4} + 16711093200 p^{5} T^{5} + 9659155 p^{10} T^{6} + 3800 p^{15} T^{7} + p^{20} T^{8}$$
19$C_2 \wr S_4$ $$1 + 3394 T + 13363997 T^{2} + 26478891042 T^{3} + 54042088934620 T^{4} + 26478891042 p^{5} T^{5} + 13363997 p^{10} T^{6} + 3394 p^{15} T^{7} + p^{20} T^{8}$$
23$C_2 \wr S_4$ $$1 + 3020 T + 9781560 T^{2} + 918428220 T^{3} + 17051675824846 T^{4} + 918428220 p^{5} T^{5} + 9781560 p^{10} T^{6} + 3020 p^{15} T^{7} + p^{20} T^{8}$$
29$C_2 \wr S_4$ $$1 + 5248 T + 39000379 T^{2} + 187477494324 T^{3} + 1321999939285496 T^{4} + 187477494324 p^{5} T^{5} + 39000379 p^{10} T^{6} + 5248 p^{15} T^{7} + p^{20} T^{8}$$
31$C_2 \wr S_4$ $$1 - 4732 T + 118023359 T^{2} - 400970802780 T^{3} + 5116274856632960 T^{4} - 400970802780 p^{5} T^{5} + 118023359 p^{10} T^{6} - 4732 p^{15} T^{7} + p^{20} T^{8}$$
37$C_2 \wr S_4$ $$1 - 10210 T + 262192965 T^{2} - 2101916693890 T^{3} + 26767783263095636 T^{4} - 2101916693890 p^{5} T^{5} + 262192965 p^{10} T^{6} - 10210 p^{15} T^{7} + p^{20} T^{8}$$
41$C_2 \wr S_4$ $$1 + 21068 T + 380218132 T^{2} + 4807939200852 T^{3} + 62995930496947958 T^{4} + 4807939200852 p^{5} T^{5} + 380218132 p^{10} T^{6} + 21068 p^{15} T^{7} + p^{20} T^{8}$$
43$C_2 \wr S_4$ $$1 + 12140 T + 451977168 T^{2} + 5413872294380 T^{3} + 90225035523150254 T^{4} + 5413872294380 p^{5} T^{5} + 451977168 p^{10} T^{6} + 12140 p^{15} T^{7} + p^{20} T^{8}$$
47$C_2 \wr S_4$ $$1 - 4720 T + 635901120 T^{2} - 2163925140240 T^{3} + 198462732407138398 T^{4} - 2163925140240 p^{5} T^{5} + 635901120 p^{10} T^{6} - 4720 p^{15} T^{7} + p^{20} T^{8}$$
53$C_2 \wr S_4$ $$1 + 21670 T + 1341191245 T^{2} + 25127588850150 T^{3} + 793079863276266116 T^{4} + 25127588850150 p^{5} T^{5} + 1341191245 p^{10} T^{6} + 21670 p^{15} T^{7} + p^{20} T^{8}$$
59$C_2 \wr S_4$ $$1 + 69068 T + 2959259616 T^{2} + 93684394769772 T^{3} + 2606995779399147790 T^{4} + 93684394769772 p^{5} T^{5} + 2959259616 p^{10} T^{6} + 69068 p^{15} T^{7} + p^{20} T^{8}$$
61$C_2 \wr S_4$ $$1 + 44000 T + 1729402923 T^{2} + 18891749550500 T^{3} + 681532725443871128 T^{4} + 18891749550500 p^{5} T^{5} + 1729402923 p^{10} T^{6} + 44000 p^{15} T^{7} + p^{20} T^{8}$$
67$C_2 \wr S_4$ $$1 + 8720 T + 3054776916 T^{2} + 68287776246800 T^{3} + 4500917017012833062 T^{4} + 68287776246800 p^{5} T^{5} + 3054776916 p^{10} T^{6} + 8720 p^{15} T^{7} + p^{20} T^{8}$$
71$C_2 \wr S_4$ $$1 + 47516 T + 4006728679 T^{2} + 40944560757036 T^{3} + 4940881051181425280 T^{4} + 40944560757036 p^{5} T^{5} + 4006728679 p^{10} T^{6} + 47516 p^{15} T^{7} + p^{20} T^{8}$$
73$C_2 \wr S_4$ $$1 + 2480 T + 2184333096 T^{2} - 56757154413520 T^{3} + 5270406230690916302 T^{4} - 56757154413520 p^{5} T^{5} + 2184333096 p^{10} T^{6} + 2480 p^{15} T^{7} + p^{20} T^{8}$$
79$C_2 \wr S_4$ $$1 + 188192 T + 20762083544 T^{2} + 1543159524876096 T^{3} + 94572488766073023566 T^{4} + 1543159524876096 p^{5} T^{5} + 20762083544 p^{10} T^{6} + 188192 p^{15} T^{7} + p^{20} T^{8}$$
83$C_2 \wr S_4$ $$1 - 68620 T + 8565816444 T^{2} - 243114964439100 T^{3} + 28749388061066642182 T^{4} - 243114964439100 p^{5} T^{5} + 8565816444 p^{10} T^{6} - 68620 p^{15} T^{7} + p^{20} T^{8}$$
89$C_2 \wr S_4$ $$1 + 170266 T + 21910171273 T^{2} + 2148925719362106 T^{3} +$$$$18\!\cdots\!88$$$$T^{4} + 2148925719362106 p^{5} T^{5} + 21910171273 p^{10} T^{6} + 170266 p^{15} T^{7} + p^{20} T^{8}$$
97$C_2 \wr S_4$ $$1 - 186160 T + 38956651512 T^{2} - 4421475549787360 T^{3} +$$$$51\!\cdots\!46$$$$T^{4} - 4421475549787360 p^{5} T^{5} + 38956651512 p^{10} T^{6} - 186160 p^{15} T^{7} + p^{20} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$