Properties

Label 4-2268e2-1.1-c0e2-0-8
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  − 7-s + 13-s + 19-s + 2·25-s + 31-s + 37-s + 43-s − 2·61-s + 67-s + 73-s + 79-s − 91-s − 2·97-s − 2·103-s + 109-s + 2·121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯
L(s)  = 1  − 7-s + 13-s + 19-s + 2·25-s + 31-s + 37-s + 43-s − 2·61-s + 67-s + 73-s + 79-s − 91-s − 2·97-s − 2·103-s + 109-s + 2·121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-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

\(L(\frac{1}{2})\) \(\approx\) \(1.319614632\)
\(L(\frac12)\) \(\approx\) \(1.319614632\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
good5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−9.317780582646917971321759921381, −9.237170542814613976888280239677, −8.512769220661477125424529172716, −8.372166272236905830619959097365, −7.898944490705169190623942359637, −7.47935974545566306832226860597, −6.88843538891503310155864051112, −6.81285161015878071365300197384, −6.19318321187977936776921390043, −6.06835495951443995566578729167, −5.49192022219242850871640485841, −5.06999682094199822997274891697, −4.53395818778021085677674268190, −4.22654243073684989423628408290, −3.46109840750289883802906301299, −3.31613638099334255020726173121, −2.75592969845313402678990414357, −2.35650938725235909753171928929, −1.30718116133893840645938333017, −0.928506364829398284787484866633, 0.928506364829398284787484866633, 1.30718116133893840645938333017, 2.35650938725235909753171928929, 2.75592969845313402678990414357, 3.31613638099334255020726173121, 3.46109840750289883802906301299, 4.22654243073684989423628408290, 4.53395818778021085677674268190, 5.06999682094199822997274891697, 5.49192022219242850871640485841, 6.06835495951443995566578729167, 6.19318321187977936776921390043, 6.81285161015878071365300197384, 6.88843538891503310155864051112, 7.47935974545566306832226860597, 7.898944490705169190623942359637, 8.372166272236905830619959097365, 8.512769220661477125424529172716, 9.237170542814613976888280239677, 9.317780582646917971321759921381

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