Properties

Label 4-2268e2-1.1-c0e2-0-10
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 − 2·13-s + 19-s + 2·25-s + 31-s − 2·37-s + 43-s + 3·49-s + 61-s − 2·67-s + 73-s − 2·79-s − 4·91-s + 97-s + 4·103-s + 109-s + 2·121-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  + 2·7-s − 2·13-s + 19-s + 2·25-s + 31-s − 2·37-s + 43-s + 3·49-s + 61-s − 2·67-s + 73-s − 2·79-s − 4·91-s + 97-s + 4·103-s + 109-s + 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯

Functional equation

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

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

\(L(\frac{1}{2})\) \(\approx\) \(1.632717920\)
\(L(\frac12)\) \(\approx\) \(1.632717920\)
\(L(1)\) 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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_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

−9.188754919180011856069537541088, −8.965197137639307565490345864460, −8.611689903288101238030312326419, −8.359206202659616191092218338091, −7.64176372835243648768349244139, −7.57190484254547160774300182641, −7.10889829460470406674706521038, −7.04600569080584849253110280983, −6.22124977003612519667269099114, −5.81265083752014871343383155148, −5.11748946445603479251160522865, −5.10522681633471481915527360708, −4.62358964249994656542382993997, −4.53575872427349756876920266939, −3.65645765918656192194199195432, −3.18668971611496039336755154062, −2.42912079975785856482785934256, −2.35274301910678028761151887825, −1.49557311344948550455527650027, −0.982146521603076594537823915266, 0.982146521603076594537823915266, 1.49557311344948550455527650027, 2.35274301910678028761151887825, 2.42912079975785856482785934256, 3.18668971611496039336755154062, 3.65645765918656192194199195432, 4.53575872427349756876920266939, 4.62358964249994656542382993997, 5.10522681633471481915527360708, 5.11748946445603479251160522865, 5.81265083752014871343383155148, 6.22124977003612519667269099114, 7.04600569080584849253110280983, 7.10889829460470406674706521038, 7.57190484254547160774300182641, 7.64176372835243648768349244139, 8.359206202659616191092218338091, 8.611689903288101238030312326419, 8.965197137639307565490345864460, 9.188754919180011856069537541088

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