Properties

Label 8-224e4-1.1-c0e4-0-1
Degree $8$
Conductor $2517630976$
Sign $1$
Analytic cond. $0.000156178$
Root an. cond. $0.334350$
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  − 16-s − 4·23-s − 4·43-s − 4·53-s + 4·67-s + 4·107-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 16-s − 4·23-s − 4·43-s − 4·53-s + 4·67-s + 4·107-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.000156178\)
Root analytic conductor: \(0.334350\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{224} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2555778200\)
\(L(\frac12)\) \(\approx\) \(0.2555778200\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−9.545982489762076035360758359071, −8.823579721935180019121213818173, −8.469677166565118245229208814431, −8.434536957138598386964682300441, −8.331101550051849999487007881591, −7.81307626176211525269068538871, −7.69871991938462507386883068052, −7.58349170248394100224433670765, −6.88124373243444318155137645877, −6.80257890134922510809578542478, −6.42454372208400273938871110723, −6.29893581866913974948321944843, −6.10004080784686032476692554731, −5.63213365356674983441617051222, −5.34425991198805266625831792784, −4.84629761488409957491405872450, −4.81090955923048536674171304725, −4.38902995657153048539593973232, −4.05555792793186239881001508243, −3.55072378156165014876097734764, −3.42045103322412422864168502433, −3.05489877579825797630944806539, −2.21337394587585197564071831515, −1.87327003141636038599724248583, −1.86659789505119545782081617086, 1.86659789505119545782081617086, 1.87327003141636038599724248583, 2.21337394587585197564071831515, 3.05489877579825797630944806539, 3.42045103322412422864168502433, 3.55072378156165014876097734764, 4.05555792793186239881001508243, 4.38902995657153048539593973232, 4.81090955923048536674171304725, 4.84629761488409957491405872450, 5.34425991198805266625831792784, 5.63213365356674983441617051222, 6.10004080784686032476692554731, 6.29893581866913974948321944843, 6.42454372208400273938871110723, 6.80257890134922510809578542478, 6.88124373243444318155137645877, 7.58349170248394100224433670765, 7.69871991938462507386883068052, 7.81307626176211525269068538871, 8.331101550051849999487007881591, 8.434536957138598386964682300441, 8.469677166565118245229208814431, 8.823579721935180019121213818173, 9.545982489762076035360758359071

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