# Properties

 Label 8-270e4-1.1-c1e4-0-1 Degree $8$ Conductor $5314410000$ Sign $1$ Analytic cond. $21.6054$ Root an. cond. $1.46831$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2·4-s + 3·16-s − 24·19-s + 9·25-s + 28·31-s − 10·49-s − 16·61-s − 4·64-s + 48·76-s − 18·100-s − 40·109-s − 6·121-s − 56·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 4-s + 3/4·16-s − 5.50·19-s + 9/5·25-s + 5.02·31-s − 1.42·49-s − 2.04·61-s − 1/2·64-s + 5.50·76-s − 9/5·100-s − 3.83·109-s − 0.545·121-s − 5.02·124-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 + 4·169-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^{4} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{12} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$21.6054$$ Root analytic conductor: $$1.46831$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7965127295$$ $$L(\frac12)$$ $$\approx$$ $$0.7965127295$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + T^{2} )^{2}$$
3 $$1$$
5$C_2^2$ $$1 - 9 T^{2} + p^{2} T^{4}$$
good7$C_2$ $$( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2}$$
11$C_2^2$ $$( 1 + 3 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 - p T^{2} )^{4}$$
17$C_2^2$ $$( 1 - 18 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
23$C_2^2$ $$( 1 - 42 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )^{4}$$
37$C_2^2$ $$( 1 + 2 T^{2} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 6 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 90 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 97 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 42 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 58 T^{2} + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{4}$$
73$C_2^2$ $$( 1 - 127 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{4}$$
83$C_2^2$ $$( 1 - 141 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 102 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 175 T^{2} + p^{2} T^{4} )^{2}$$
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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

−8.525631392692962727035906310322, −8.455040859108733109926187603313, −8.351675457227233897594458238514, −8.053229304902307451834733415791, −7.74882062897673679131828797956, −7.68264471490111978070905826614, −6.74297340677452860664192877079, −6.70548239590063049571171499498, −6.63369949335041670626880174907, −6.31807200379814711901959686957, −6.27720123821195438392519419345, −5.83624473813148852273036815983, −5.39705184573516012405240074521, −4.83113162903713253478382194672, −4.73120116441461174560227768954, −4.63088320330289034880030398323, −4.25958593489423691737548826188, −4.04094593557720861627866626263, −3.79686094233131321407077484596, −2.90956297276339527698007690151, −2.77550267018518104598069753251, −2.61509841044716268569206278134, −1.89475710595397661077940748203, −1.39479614281458716766363622217, −0.46931538822514545683082698665, 0.46931538822514545683082698665, 1.39479614281458716766363622217, 1.89475710595397661077940748203, 2.61509841044716268569206278134, 2.77550267018518104598069753251, 2.90956297276339527698007690151, 3.79686094233131321407077484596, 4.04094593557720861627866626263, 4.25958593489423691737548826188, 4.63088320330289034880030398323, 4.73120116441461174560227768954, 4.83113162903713253478382194672, 5.39705184573516012405240074521, 5.83624473813148852273036815983, 6.27720123821195438392519419345, 6.31807200379814711901959686957, 6.63369949335041670626880174907, 6.70548239590063049571171499498, 6.74297340677452860664192877079, 7.68264471490111978070905826614, 7.74882062897673679131828797956, 8.053229304902307451834733415791, 8.351675457227233897594458238514, 8.455040859108733109926187603313, 8.525631392692962727035906310322