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Properties

Label 8-3332e4-1.1-c0e4-0-0
Degree $8$
Conductor $1.233\times 10^{14}$
Sign $1$
Analytic cond. $7.64624$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 16-s + 10·25-s − 4·37-s − 4·53-s − 4·73-s + 4·80-s − 4·97-s + 4·109-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 16·185-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 4·5-s − 16-s + 10·25-s − 4·37-s − 4·53-s − 4·73-s + 4·80-s − 4·97-s + 4·109-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 16·185-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 7^{8} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(7.64624\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 7^{8} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006397139721\)
\(L(\frac12)\) \(\approx\) \(0.006397139721\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
7 \( 1 \)
17$C_2^2$ \( 1 + T^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
41$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
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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

−6.36371574000492539707924882567, −6.35434157824868879582503173310, −5.87900314091183779137229109132, −5.57225774658692846766056837693, −5.48542224060844217595885183287, −5.18765683861963310014032409969, −4.79321831044558780763752905131, −4.73983473518180413407738537025, −4.69926971905537516192390360629, −4.51232389073341059948598171692, −4.34543523930817536884988244333, −3.88858241692880972587115339182, −3.83606717947741136732340941936, −3.67753770248503293400779734521, −3.29677126391270769567097625217, −3.25036334189636790909054403180, −3.19974757538479817229075309760, −2.77768634962451064038636191207, −2.56217090576691343655968275193, −2.31257225325893339478800050740, −1.53677249224403310468810495226, −1.51765393225396190521489993603, −1.46142493655953190919526617296, −0.67462249582215132399735969443, −0.04569502695715513764472224181, 0.04569502695715513764472224181, 0.67462249582215132399735969443, 1.46142493655953190919526617296, 1.51765393225396190521489993603, 1.53677249224403310468810495226, 2.31257225325893339478800050740, 2.56217090576691343655968275193, 2.77768634962451064038636191207, 3.19974757538479817229075309760, 3.25036334189636790909054403180, 3.29677126391270769567097625217, 3.67753770248503293400779734521, 3.83606717947741136732340941936, 3.88858241692880972587115339182, 4.34543523930817536884988244333, 4.51232389073341059948598171692, 4.69926971905537516192390360629, 4.73983473518180413407738537025, 4.79321831044558780763752905131, 5.18765683861963310014032409969, 5.48542224060844217595885183287, 5.57225774658692846766056837693, 5.87900314091183779137229109132, 6.35434157824868879582503173310, 6.36371574000492539707924882567

Graph of the $Z$-function along the critical line