# Properties

 Label 8-96e8-1.1-c1e4-0-16 Degree $8$ Conductor $7.214\times 10^{15}$ Sign $1$ Analytic cond. $2.93277\times 10^{7}$ Root an. cond. $8.57846$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 12·11-s + 12·19-s − 8·25-s + 12·43-s − 12·49-s − 36·59-s + 4·67-s − 24·73-s − 12·83-s + 24·89-s − 12·107-s + 24·113-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 3.61·11-s + 2.75·19-s − 8/5·25-s + 1.82·43-s − 1.71·49-s − 4.68·59-s + 0.488·67-s − 2.80·73-s − 1.31·83-s + 2.54·89-s − 1.16·107-s + 2.25·113-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{40} \cdot 3^{8}$$ Sign: $1$ Analytic conductor: $$2.93277\times 10^{7}$$ Root analytic conductor: $$8.57846$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
3 $$1$$
good5$C_2^2$ $$( 1 + 4 T^{2} + p^{2} T^{4} )^{2}$$
7$D_4\times C_2$ $$1 + 12 T^{2} + 86 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 + 24 T^{2} + 290 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2^2$ $$( 1 + 22 T^{2} + p^{2} T^{4} )^{2}$$
19$D_{4}$ $$( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 + 44 T^{2} + 1110 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 + 52 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 30 T^{2} + p^{2} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 + 120 T^{2} + 6146 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2^2$ $$( 1 + 34 T^{2} + p^{2} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 + 56 T^{2} + 4674 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 + 216 T^{2} + 18914 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 + 236 T^{2} + 23574 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8}$$
73$D_{4}$ $$( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 + 60 T^{2} + 1094 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_{4}$ $$( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 + 182 T^{2} + p^{2} T^{4} )^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−5.76586023163380844735023261844, −5.56305602401833478554533887826, −5.41340794334289390807313225086, −5.17083454201060076641059943229, −5.06584579977950825697189783505, −4.84483218690414016283951175678, −4.75691052501699095455252087629, −4.60656649933578785513102597679, −4.57561431435825007113081002319, −4.00795793533278256012302000519, −3.81019954474159096782987636640, −3.79026564454694821170095185796, −3.55897407453321397963713090078, −3.17829072932933740725510268348, −3.01362713234189801599954230886, −2.97866686790889399500362693118, −2.88522781551283893534036070942, −2.44171206121185618103970598907, −2.38717062268212557857373040493, −2.14635524754993739289785962849, −2.06819093831957746679997156532, −1.44465008667432960132642291136, −1.22020415911959115229984470536, −1.20671576777011779905573360833, −1.05140726223231886858073163856, 0, 0, 0, 0, 1.05140726223231886858073163856, 1.20671576777011779905573360833, 1.22020415911959115229984470536, 1.44465008667432960132642291136, 2.06819093831957746679997156532, 2.14635524754993739289785962849, 2.38717062268212557857373040493, 2.44171206121185618103970598907, 2.88522781551283893534036070942, 2.97866686790889399500362693118, 3.01362713234189801599954230886, 3.17829072932933740725510268348, 3.55897407453321397963713090078, 3.79026564454694821170095185796, 3.81019954474159096782987636640, 4.00795793533278256012302000519, 4.57561431435825007113081002319, 4.60656649933578785513102597679, 4.75691052501699095455252087629, 4.84483218690414016283951175678, 5.06584579977950825697189783505, 5.17083454201060076641059943229, 5.41340794334289390807313225086, 5.56305602401833478554533887826, 5.76586023163380844735023261844