Properties

Label 4-6e8-1.1-c0e2-0-1
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $0.418335$
Root an. cond. $0.804231$
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  − 7-s + 13-s + 2·19-s − 25-s + 2·31-s − 2·37-s + 2·43-s + 49-s + 61-s − 67-s − 2·73-s − 79-s − 91-s + 97-s − 103-s + 4·109-s − 121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 7-s + 13-s + 2·19-s − 25-s + 2·31-s − 2·37-s + 2·43-s + 49-s + 61-s − 67-s − 2·73-s − 79-s − 91-s + 97-s − 103-s + 4·109-s − 121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(0.418335\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1679616,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034688834\)
\(L(\frac12)\) \(\approx\) \(1.034688834\)
\(L(1)\) 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 \( 1 \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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

−9.976460439726058876802478941422, −9.857554802837019139384160546371, −9.056943556033875290193898870540, −8.968348658327949297262650249439, −8.607493425990316238140279698133, −7.960091180218141000908313520896, −7.43021332229877915758758437671, −7.42429584257468520538659864693, −6.67852976714919991226496073411, −6.37565511981828672964346474963, −5.78394001834080959642183209820, −5.69412703741520531046856888022, −5.02757464026046468759624831698, −4.50009399754335994416246096190, −3.79245552020892662679042162058, −3.64716782261660270103531063276, −2.90003139280413342578176578766, −2.65918547005904147011194791027, −1.62883214494859894648940785575, −0.956525500722701893500926710313, 0.956525500722701893500926710313, 1.62883214494859894648940785575, 2.65918547005904147011194791027, 2.90003139280413342578176578766, 3.64716782261660270103531063276, 3.79245552020892662679042162058, 4.50009399754335994416246096190, 5.02757464026046468759624831698, 5.69412703741520531046856888022, 5.78394001834080959642183209820, 6.37565511981828672964346474963, 6.67852976714919991226496073411, 7.42429584257468520538659864693, 7.43021332229877915758758437671, 7.960091180218141000908313520896, 8.607493425990316238140279698133, 8.968348658327949297262650249439, 9.056943556033875290193898870540, 9.857554802837019139384160546371, 9.976460439726058876802478941422

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