Properties

Label 8-2880e4-1.1-c0e4-0-1
Degree $8$
Conductor $6.880\times 10^{13}$
Sign $1$
Analytic cond. $4.26774$
Root an. cond. $1.19887$
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  − 2·5-s + 9-s + 25-s − 2·29-s − 2·41-s − 2·45-s − 49-s + 2·61-s − 4·89-s + 4·101-s − 4·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·5-s + 9-s + 25-s − 2·29-s − 2·41-s − 2·45-s − 49-s + 2·61-s − 4·89-s + 4·101-s − 4·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(4.26774\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2717722466\)
\(L(\frac12)\) \(\approx\) \(0.2717722466\)
\(L(1)\) 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 \( 1 \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
good7$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
89$C_2$ \( ( 1 + T + T^{2} )^{4} \)
97$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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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

−6.33438963666772706290111264723, −6.27479980721362033906995617869, −6.13955074586187396378085917121, −5.58718804853802737488813131175, −5.53277296425893229032705217626, −5.51988635997385261160774587765, −5.12352981424206365054936397851, −5.01900748094432207229325186479, −4.67820127715461474887788639533, −4.47822221018584133888104016738, −4.23659397101326177822823679224, −4.09940668331711756428119696595, −3.92548777695194507721642050241, −3.79362776807097909848194366055, −3.42716770290077940993653432234, −3.27236108783644115050752032575, −3.24754689870159944293772371394, −2.66862849851516393678702060796, −2.62351533429455454109864376841, −1.99795644573682382093650688303, −1.97488064813586656689865332117, −1.70702938926254345453074260019, −1.22300475139456811455549182232, −1.05167309494442332897635701643, −0.23598449708717053306657858830, 0.23598449708717053306657858830, 1.05167309494442332897635701643, 1.22300475139456811455549182232, 1.70702938926254345453074260019, 1.97488064813586656689865332117, 1.99795644573682382093650688303, 2.62351533429455454109864376841, 2.66862849851516393678702060796, 3.24754689870159944293772371394, 3.27236108783644115050752032575, 3.42716770290077940993653432234, 3.79362776807097909848194366055, 3.92548777695194507721642050241, 4.09940668331711756428119696595, 4.23659397101326177822823679224, 4.47822221018584133888104016738, 4.67820127715461474887788639533, 5.01900748094432207229325186479, 5.12352981424206365054936397851, 5.51988635997385261160774587765, 5.53277296425893229032705217626, 5.58718804853802737488813131175, 6.13955074586187396378085917121, 6.27479980721362033906995617869, 6.33438963666772706290111264723

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