Properties

Label 8-1216e4-1.1-c0e4-0-0
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $0.135632$
Root an. cond. $0.779014$
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  − 4·9-s − 4·17-s − 2·25-s + 2·49-s + 4·73-s + 10·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4·9-s − 4·17-s − 2·25-s + 2·49-s + 4·73-s + 10·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(0.135632\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1718022968\)
\(L(\frac12)\) \(\approx\) \(0.1718022968\)
\(L(1)\) 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 \( 1 \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2$ \( ( 1 + T^{2} )^{4} \)
5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_2$ \( ( 1 + T + T^{2} )^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 - T + T^{2} )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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

−7.02422408917260611142478651111, −6.95133679846000348904702588559, −6.64934749368689905267604189069, −6.37512050132226794038680372742, −6.21420698267797631279798655882, −6.10944500377023084003472926039, −5.96936623512845364835318545403, −5.65278155660643339239987732125, −5.37173359460988602407981346933, −5.21404517380726172785640661391, −4.97994506259223041681830704473, −4.77758906319748045592440684855, −4.50102483581328217115477055380, −4.05939879201996381022235652347, −3.86138599817062274226591239329, −3.80143317610510809605380709422, −3.36803280152745760360360494033, −3.15437356169946264646723041268, −2.68253133036454636010506112612, −2.54668252330011589842346845765, −2.18897370204960924819179293151, −2.14279730663332803716379781987, −2.08601991965922016682182788152, −1.06166890326627544015399119234, −0.29217866896677768525668913622, 0.29217866896677768525668913622, 1.06166890326627544015399119234, 2.08601991965922016682182788152, 2.14279730663332803716379781987, 2.18897370204960924819179293151, 2.54668252330011589842346845765, 2.68253133036454636010506112612, 3.15437356169946264646723041268, 3.36803280152745760360360494033, 3.80143317610510809605380709422, 3.86138599817062274226591239329, 4.05939879201996381022235652347, 4.50102483581328217115477055380, 4.77758906319748045592440684855, 4.97994506259223041681830704473, 5.21404517380726172785640661391, 5.37173359460988602407981346933, 5.65278155660643339239987732125, 5.96936623512845364835318545403, 6.10944500377023084003472926039, 6.21420698267797631279798655882, 6.37512050132226794038680372742, 6.64934749368689905267604189069, 6.95133679846000348904702588559, 7.02422408917260611142478651111

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