Properties

Label 4-2268e2-1.1-c0e2-0-6
Degree $4$
Conductor $5143824$
Sign $1$
Analytic cond. $1.28115$
Root an. cond. $1.06389$
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  − 2·7-s + 3·13-s − 25-s + 2·37-s + 2·43-s + 3·49-s + 3·61-s + 67-s + 79-s − 6·91-s − 3·97-s − 3·103-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s + 2·175-s + 179-s + ⋯
L(s)  = 1  − 2·7-s + 3·13-s − 25-s + 2·37-s + 2·43-s + 3·49-s + 3·61-s + 67-s + 79-s − 6·91-s − 3·97-s − 3·103-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s + 2·175-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.28115\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2268} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5143824,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173480047\)
\(L(\frac12)\) \(\approx\) \(1.173480047\)
\(L(1)\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.490457717169550062306459628965, −9.086904130374747699671617419812, −8.515269951709445671315061169925, −8.424365657196634871642871830394, −7.942301066710654382239133360488, −7.49380280766011637266041688068, −6.79195093489357723926246348362, −6.73913834344124636862697925753, −6.23624959591087673103876090960, −5.98068878866284274193485710456, −5.60207432715125044375550320083, −5.34308692405459366573297979153, −4.14613535989302176557514583390, −4.10244163824102482682974948997, −3.75565811128662612243217467035, −3.36232265057665199431457959435, −2.59468350929736852745878501879, −2.47277485950077349043433255441, −1.32527662649462541694657461225, −0.851976924933049849242585471126, 0.851976924933049849242585471126, 1.32527662649462541694657461225, 2.47277485950077349043433255441, 2.59468350929736852745878501879, 3.36232265057665199431457959435, 3.75565811128662612243217467035, 4.10244163824102482682974948997, 4.14613535989302176557514583390, 5.34308692405459366573297979153, 5.60207432715125044375550320083, 5.98068878866284274193485710456, 6.23624959591087673103876090960, 6.73913834344124636862697925753, 6.79195093489357723926246348362, 7.49380280766011637266041688068, 7.942301066710654382239133360488, 8.424365657196634871642871830394, 8.515269951709445671315061169925, 9.086904130374747699671617419812, 9.490457717169550062306459628965

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