# Properties

 Label 8-74e4-1.1-c7e4-0-0 Degree $8$ Conductor $29986576$ Sign $1$ Analytic cond. $285553.$ Root an. cond. $4.80796$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 32·2-s − 53·3-s + 640·4-s + 111·5-s + 1.69e3·6-s − 1.66e3·7-s − 1.02e4·8-s − 665·9-s − 3.55e3·10-s − 4.59e3·11-s − 3.39e4·12-s + 7.84e3·13-s + 5.33e4·14-s − 5.88e3·15-s + 1.43e5·16-s + 2.31e4·17-s + 2.12e4·18-s + 2.36e4·19-s + 7.10e4·20-s + 8.82e4·21-s + 1.46e5·22-s + 2.41e4·23-s + 5.42e5·24-s − 2.19e5·25-s − 2.51e5·26-s − 2.10e4·27-s − 1.06e6·28-s + ⋯
 L(s)  = 1 − 2.82·2-s − 1.13·3-s + 5·4-s + 0.397·5-s + 3.20·6-s − 1.83·7-s − 7.07·8-s − 0.304·9-s − 1.12·10-s − 1.04·11-s − 5.66·12-s + 0.990·13-s + 5.19·14-s − 0.450·15-s + 35/4·16-s + 1.14·17-s + 0.860·18-s + 0.792·19-s + 1.98·20-s + 2.08·21-s + 2.94·22-s + 0.413·23-s + 8.01·24-s − 2.80·25-s − 2.80·26-s − 0.205·27-s − 9.17·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$29986576$$    =    $$2^{4} \cdot 37^{4}$$ Sign: $1$ Analytic conductor: $$285553.$$ Root analytic conductor: $$4.80796$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 29986576,\ (\ :7/2, 7/2, 7/2, 7/2),\ 1)$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ 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$C_1$ $$( 1 + p^{3} T )^{4}$$
37$C_1$ $$( 1 - p^{3} T )^{4}$$
good3$C_2 \wr S_4$ $$1 + 53 T + 386 p^{2} T^{2} + 2968 p^{4} T^{3} + 348569 p^{3} T^{4} + 2968 p^{11} T^{5} + 386 p^{16} T^{6} + 53 p^{21} T^{7} + p^{28} T^{8}$$
5$C_2 \wr S_4$ $$1 - 111 T + 46297 p T^{2} - 1302973 p^{2} T^{3} + 190908864 p^{3} T^{4} - 1302973 p^{9} T^{5} + 46297 p^{15} T^{6} - 111 p^{21} T^{7} + p^{28} T^{8}$$
7$C_2 \wr S_4$ $$1 + 34 p^{2} T + 3836251 T^{2} + 3980351580 T^{3} + 4954084913556 T^{4} + 3980351580 p^{7} T^{5} + 3836251 p^{14} T^{6} + 34 p^{23} T^{7} + p^{28} T^{8}$$
11$C_2 \wr S_4$ $$1 + 4593 T + 58690784 T^{2} + 139324724144 T^{3} + 1358111660813121 T^{4} + 139324724144 p^{7} T^{5} + 58690784 p^{14} T^{6} + 4593 p^{21} T^{7} + p^{28} T^{8}$$
13$C_2 \wr S_4$ $$1 - 7847 T + 44905309 T^{2} + 238876335675 T^{3} - 5523621661124304 T^{4} + 238876335675 p^{7} T^{5} + 44905309 p^{14} T^{6} - 7847 p^{21} T^{7} + p^{28} T^{8}$$
17$C_2 \wr S_4$ $$1 - 23172 T + 1565149604 T^{2} - 26309358229308 T^{3} + 956973477754784886 T^{4} - 26309358229308 p^{7} T^{5} + 1565149604 p^{14} T^{6} - 23172 p^{21} T^{7} + p^{28} T^{8}$$
19$C_2 \wr S_4$ $$1 - 23696 T + 2348541316 T^{2} - 44830471203152 T^{3} + 2917224547808407606 T^{4} - 44830471203152 p^{7} T^{5} + 2348541316 p^{14} T^{6} - 23696 p^{21} T^{7} + p^{28} T^{8}$$
23$C_2 \wr S_4$ $$1 - 24105 T + 7013385317 T^{2} - 88359223748697 T^{3} + 29217779771576109072 T^{4} - 88359223748697 p^{7} T^{5} + 7013385317 p^{14} T^{6} - 24105 p^{21} T^{7} + p^{28} T^{8}$$
29$C_2 \wr S_4$ $$1 + 140949 T + 69012666881 T^{2} + 6785207860407915 T^{3} +$$$$17\!\cdots\!36$$$$T^{4} + 6785207860407915 p^{7} T^{5} + 69012666881 p^{14} T^{6} + 140949 p^{21} T^{7} + p^{28} T^{8}$$
31$C_2 \wr S_4$ $$1 + 21439 p T + 256765050331 T^{2} + 67005695462988077 T^{3} +$$$$12\!\cdots\!88$$$$T^{4} + 67005695462988077 p^{7} T^{5} + 256765050331 p^{14} T^{6} + 21439 p^{22} T^{7} + p^{28} T^{8}$$
41$C_2 \wr S_4$ $$1 + 709737 T + 268358061962 T^{2} + 179477185301479320 T^{3} +$$$$11\!\cdots\!17$$$$T^{4} + 179477185301479320 p^{7} T^{5} + 268358061962 p^{14} T^{6} + 709737 p^{21} T^{7} + p^{28} T^{8}$$
43$C_2 \wr S_4$ $$1 + 128962 T + 680217086800 T^{2} + 72403133514238178 T^{3} +$$$$24\!\cdots\!46$$$$T^{4} + 72403133514238178 p^{7} T^{5} + 680217086800 p^{14} T^{6} + 128962 p^{21} T^{7} + p^{28} T^{8}$$
47$C_2 \wr S_4$ $$1 + 9486 p T + 1135342318835 T^{2} + 614279906235197180 T^{3} +$$$$71\!\cdots\!84$$$$T^{4} + 614279906235197180 p^{7} T^{5} + 1135342318835 p^{14} T^{6} + 9486 p^{22} T^{7} + p^{28} T^{8}$$
53$C_2 \wr S_4$ $$1 + 975870 T + 2112432693629 T^{2} - 517582713481480554 T^{3} +$$$$79\!\cdots\!80$$$$T^{4} - 517582713481480554 p^{7} T^{5} + 2112432693629 p^{14} T^{6} + 975870 p^{21} T^{7} + p^{28} T^{8}$$
59$C_2 \wr S_4$ $$1 + 1812858 T + 9776352637904 T^{2} + 11765178797880874346 T^{3} +$$$$35\!\cdots\!58$$$$T^{4} + 11765178797880874346 p^{7} T^{5} + 9776352637904 p^{14} T^{6} + 1812858 p^{21} T^{7} + p^{28} T^{8}$$
61$C_2 \wr S_4$ $$1 + 2955031 T + 10161069366493 T^{2} + 20511634928342419193 T^{3} +$$$$40\!\cdots\!24$$$$T^{4} + 20511634928342419193 p^{7} T^{5} + 10161069366493 p^{14} T^{6} + 2955031 p^{21} T^{7} + p^{28} T^{8}$$
67$C_2 \wr S_4$ $$1 - 2737235 T + 11338696552561 T^{2} - 10253324599054769427 T^{3} +$$$$42\!\cdots\!00$$$$T^{4} - 10253324599054769427 p^{7} T^{5} + 11338696552561 p^{14} T^{6} - 2737235 p^{21} T^{7} + p^{28} T^{8}$$
71$C_2 \wr S_4$ $$1 - 4958184 T + 34530824802083 T^{2} -$$$$13\!\cdots\!88$$$$T^{3} +$$$$46\!\cdots\!44$$$$T^{4} -$$$$13\!\cdots\!88$$$$p^{7} T^{5} + 34530824802083 p^{14} T^{6} - 4958184 p^{21} T^{7} + p^{28} T^{8}$$
73$C_2 \wr S_4$ $$1 + 931591 T + 36783900926266 T^{2} + 23785151910851054960 T^{3} +$$$$57\!\cdots\!97$$$$T^{4} + 23785151910851054960 p^{7} T^{5} + 36783900926266 p^{14} T^{6} + 931591 p^{21} T^{7} + p^{28} T^{8}$$
79$C_2 \wr S_4$ $$1 - 5813561 T + 62570152478953 T^{2} -$$$$28\!\cdots\!17$$$$T^{3} +$$$$17\!\cdots\!80$$$$T^{4} -$$$$28\!\cdots\!17$$$$p^{7} T^{5} + 62570152478953 p^{14} T^{6} - 5813561 p^{21} T^{7} + p^{28} T^{8}$$
83$C_2 \wr S_4$ $$1 - 2120460 T + 82604185739147 T^{2} -$$$$14\!\cdots\!40$$$$T^{3} +$$$$30\!\cdots\!00$$$$T^{4} -$$$$14\!\cdots\!40$$$$p^{7} T^{5} + 82604185739147 p^{14} T^{6} - 2120460 p^{21} T^{7} + p^{28} T^{8}$$
89$C_2 \wr S_4$ $$1 - 8833716 T + 177304580185760 T^{2} -$$$$11\!\cdots\!56$$$$T^{3} +$$$$11\!\cdots\!98$$$$T^{4} -$$$$11\!\cdots\!56$$$$p^{7} T^{5} + 177304580185760 p^{14} T^{6} - 8833716 p^{21} T^{7} + p^{28} T^{8}$$
97$C_2 \wr S_4$ $$1 + 22666876 T + 479752420528504 T^{2} +$$$$58\!\cdots\!32$$$$T^{3} +$$$$64\!\cdots\!86$$$$T^{4} +$$$$58\!\cdots\!32$$$$p^{7} T^{5} + 479752420528504 p^{14} T^{6} + 22666876 p^{21} T^{7} + p^{28} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$