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Properties

Label 8-3060e4-1.1-c0e4-0-6
Degree $8$
Conductor $8.768\times 10^{13}$
Sign $1$
Analytic cond. $5.43893$
Root an. cond. $1.23577$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 4·5-s + 3·16-s + 8·20-s + 10·25-s + 4·37-s − 4·41-s + 4·61-s − 4·64-s − 12·80-s − 20·100-s + 4·113-s − 20·125-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 8·164-s + 167-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·4-s − 4·5-s + 3·16-s + 8·20-s + 10·25-s + 4·37-s − 4·41-s + 4·61-s − 4·64-s − 12·80-s − 20·100-s + 4·113-s − 20·125-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 8·164-s + 167-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(5.43893\)
Root analytic conductor: \(1.23577\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2987711612\)
\(L(\frac12)\) \(\approx\) \(0.2987711612\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
good7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
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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

−6.40432356096246403988710536498, −6.18920785490372624488810875412, −6.05103511084786689468852384857, −5.55885204630941910387505606096, −5.34517008972264058414951655678, −5.19404830518418267265084056944, −5.14758909093696821247162837715, −4.77806414895476646089563619017, −4.52277486078829426413255940817, −4.49367306557534584681727231227, −4.40159948885944088567112804343, −4.10032987503795098962753171779, −3.76569470553966414620332014040, −3.75952118783003240219820991293, −3.65546508223863718540567460531, −3.22391913276508555832504986243, −3.15928419651365065528078411306, −3.00880113309762796933618804269, −2.63272038073415523520032348208, −2.27878364868176899448225987784, −1.85947190581047881209076048247, −1.33192172199807774396006871156, −0.928068737235821413835795941875, −0.74215609056171383761969078360, −0.44390444102033722345138182037, 0.44390444102033722345138182037, 0.74215609056171383761969078360, 0.928068737235821413835795941875, 1.33192172199807774396006871156, 1.85947190581047881209076048247, 2.27878364868176899448225987784, 2.63272038073415523520032348208, 3.00880113309762796933618804269, 3.15928419651365065528078411306, 3.22391913276508555832504986243, 3.65546508223863718540567460531, 3.75952118783003240219820991293, 3.76569470553966414620332014040, 4.10032987503795098962753171779, 4.40159948885944088567112804343, 4.49367306557534584681727231227, 4.52277486078829426413255940817, 4.77806414895476646089563619017, 5.14758909093696821247162837715, 5.19404830518418267265084056944, 5.34517008972264058414951655678, 5.55885204630941910387505606096, 6.05103511084786689468852384857, 6.18920785490372624488810875412, 6.40432356096246403988710536498

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