# Properties

 Label 8-740e4-1.1-c0e4-0-0 Degree $8$ Conductor $299865760000$ Sign $1$ Analytic cond. $0.0186018$ Root an. cond. $0.607707$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 2·17-s − 2·25-s + 2·29-s + 6·41-s − 2·53-s − 4·61-s − 64-s + 2·68-s − 4·73-s + 81-s − 2·89-s − 2·100-s + 2·109-s − 4·113-s + 2·116-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + 167-s + ⋯
 L(s)  = 1 + 4-s + 2·17-s − 2·25-s + 2·29-s + 6·41-s − 2·53-s − 4·61-s − 64-s + 2·68-s − 4·73-s + 81-s − 2·89-s − 2·100-s + 2·109-s − 4·113-s + 2·116-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + 167-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 5^{4} \cdot 37^{4}$$ Sign: $1$ Analytic conductor: $$0.0186018$$ Root analytic conductor: $$0.607707$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.022283297$$ $$L(\frac12)$$ $$\approx$$ $$1.022283297$$ $$L(1)$$ 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$C_2^2$ $$1 - T^{2} + T^{4}$$
5$C_2$ $$( 1 + T^{2} )^{2}$$
37$C_2^2$ $$1 - T^{2} + T^{4}$$
good3$C_2^3$ $$1 - T^{4} + T^{8}$$
7$C_2^3$ $$1 - T^{4} + T^{8}$$
11$C_2$ $$( 1 + T^{2} )^{4}$$
13$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
17$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
19$C_2^3$ $$1 - T^{4} + T^{8}$$
23$C_2$ $$( 1 + T^{2} )^{4}$$
29$C_2$$\times$$C_2^2$ $$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
31$C_2^2$ $$( 1 + T^{4} )^{2}$$
41$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 - T + T^{2} )^{2}$$
43$C_2$ $$( 1 + T^{2} )^{4}$$
47$C_2^2$ $$( 1 + T^{4} )^{2}$$
53$C_2$$\times$$C_2^2$ $$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
59$C_2^3$ $$1 - T^{4} + T^{8}$$
61$C_1$$\times$$C_2^2$ $$( 1 + T )^{4}( 1 - T^{2} + T^{4} )$$
67$C_2^3$ $$1 - T^{4} + T^{8}$$
71$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
73$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 + T^{2} )^{2}$$
79$C_2^3$ $$1 - T^{4} + T^{8}$$
83$C_2^3$ $$1 - T^{4} + T^{8}$$
89$C_2$$\times$$C_2$ $$( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
97$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
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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

−7.76335069770711380815944705751, −7.48802752535027932945038149750, −7.36354870971581762264830516389, −7.13405408471555076256760807172, −6.59045170453058589149057285217, −6.46479103134850073221741867070, −6.27187084497638680750546371118, −6.07679696028216140005878011853, −5.84606518034813986154252965710, −5.61800805877455403063854783559, −5.60499459741940231579639292109, −5.00090472267369864288036427568, −4.81917691633289795937113065316, −4.38192199185416200915025067144, −4.35728177321846310550292871526, −3.96896136100766709259412529670, −3.91098326614564103591735517081, −3.24791354945668227586490716986, −3.03133711333930172747107426393, −2.80144763278416741010262044482, −2.55604529841625931071427480001, −2.42304464500207117343682442367, −1.59090044178183410278437293239, −1.44415463942121210603204209397, −1.15443793449386252972175086115, 1.15443793449386252972175086115, 1.44415463942121210603204209397, 1.59090044178183410278437293239, 2.42304464500207117343682442367, 2.55604529841625931071427480001, 2.80144763278416741010262044482, 3.03133711333930172747107426393, 3.24791354945668227586490716986, 3.91098326614564103591735517081, 3.96896136100766709259412529670, 4.35728177321846310550292871526, 4.38192199185416200915025067144, 4.81917691633289795937113065316, 5.00090472267369864288036427568, 5.60499459741940231579639292109, 5.61800805877455403063854783559, 5.84606518034813986154252965710, 6.07679696028216140005878011853, 6.27187084497638680750546371118, 6.46479103134850073221741867070, 6.59045170453058589149057285217, 7.13405408471555076256760807172, 7.36354870971581762264830516389, 7.48802752535027932945038149750, 7.76335069770711380815944705751