Properties

Label 4-2268e2-1.1-c1e2-0-8
Degree $4$
Conductor $5143824$
Sign $1$
Analytic cond. $327.974$
Root an. cond. $4.25559$
Motivic weight $1$
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  − 4·7-s − 2·13-s + 19-s − 10·25-s + 7·31-s + 10·37-s − 5·43-s + 9·49-s + 61-s + 16·67-s − 17·73-s + 4·79-s + 8·91-s + 19·97-s + 40·103-s − 17·109-s − 22·121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.554·13-s + 0.229·19-s − 2·25-s + 1.25·31-s + 1.64·37-s − 0.762·43-s + 9/7·49-s + 0.128·61-s + 1.95·67-s − 1.98·73-s + 0.450·79-s + 0.838·91-s + 1.92·97-s + 3.94·103-s − 1.62·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 0.346·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(327.974\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2268} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5143824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.015417720\)
\(L(\frac12)\) \(\approx\) \(1.015417720\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
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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

−9.291122328611718349895599706417, −8.958011323934146674340728859508, −8.425267446671817411179928971512, −8.007623087298332746234245775671, −7.61218400205677995047146541298, −7.40088258895552779700720193226, −6.82927548833839094404464875102, −6.30817956212715710106039443967, −6.25246162473715572955443888923, −5.85281977791787056277457947149, −5.22556683155357210348381521104, −4.89847339124623733294036718634, −4.28581574043032567823460635352, −3.87201055401858139275699630868, −3.51342737769171320840106795590, −2.93978152154037080513733695321, −2.51000669199540901406570886824, −2.06653202313522153868653093689, −1.16353929674270346245792418559, −0.37259392282041021892127344074, 0.37259392282041021892127344074, 1.16353929674270346245792418559, 2.06653202313522153868653093689, 2.51000669199540901406570886824, 2.93978152154037080513733695321, 3.51342737769171320840106795590, 3.87201055401858139275699630868, 4.28581574043032567823460635352, 4.89847339124623733294036718634, 5.22556683155357210348381521104, 5.85281977791787056277457947149, 6.25246162473715572955443888923, 6.30817956212715710106039443967, 6.82927548833839094404464875102, 7.40088258895552779700720193226, 7.61218400205677995047146541298, 8.007623087298332746234245775671, 8.425267446671817411179928971512, 8.958011323934146674340728859508, 9.291122328611718349895599706417

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