Properties

Label 4-162e2-1.1-c1e2-0-1
Degree $4$
Conductor $26244$
Sign $1$
Analytic cond. $1.67334$
Root an. cond. $1.13735$
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  + 2-s − 3·5-s + 7-s − 8-s − 3·10-s + 3·11-s + 4·13-s + 14-s − 16-s + 4·19-s + 3·22-s + 6·23-s + 5·25-s + 4·26-s − 6·29-s − 5·31-s − 3·35-s + 4·37-s + 4·38-s + 3·40-s + 6·41-s + 10·43-s + 6·46-s − 6·47-s + 7·49-s + 5·50-s + 18·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.34·5-s + 0.377·7-s − 0.353·8-s − 0.948·10-s + 0.904·11-s + 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.917·19-s + 0.639·22-s + 1.25·23-s + 25-s + 0.784·26-s − 1.11·29-s − 0.898·31-s − 0.507·35-s + 0.657·37-s + 0.648·38-s + 0.474·40-s + 0.937·41-s + 1.52·43-s + 0.884·46-s − 0.875·47-s + 49-s + 0.707·50-s + 2.47·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26244\)    =    \(2^{2} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.67334\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{162} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 26244,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.476621706\)
\(L(\frac12)\) \(\approx\) \(1.476621706\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−13.20502644880271505762667057528, −12.52957867937897606301988860266, −12.19283051970361002428092658245, −11.56723079676035538687377000221, −11.23613223905438751972022657379, −11.01121290786422321051583622792, −10.25053277570361930924207602017, −9.372554760033261947426602530755, −8.834936174781593452814909941403, −8.751635463279047721042983295697, −7.66905073689947661154738095313, −7.44846691449250731958547779257, −6.90330704735414953485190453403, −5.81731870596322255653774703433, −5.71641139191517235358585726381, −4.56099257260541310401616603002, −4.20445647602309845509273962127, −3.59206387325909758127408916542, −2.91754008419131711827962264885, −1.24707914124000999913542719082, 1.24707914124000999913542719082, 2.91754008419131711827962264885, 3.59206387325909758127408916542, 4.20445647602309845509273962127, 4.56099257260541310401616603002, 5.71641139191517235358585726381, 5.81731870596322255653774703433, 6.90330704735414953485190453403, 7.44846691449250731958547779257, 7.66905073689947661154738095313, 8.751635463279047721042983295697, 8.834936174781593452814909941403, 9.372554760033261947426602530755, 10.25053277570361930924207602017, 11.01121290786422321051583622792, 11.23613223905438751972022657379, 11.56723079676035538687377000221, 12.19283051970361002428092658245, 12.52957867937897606301988860266, 13.20502644880271505762667057528

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