Properties

Label 4-1142784-1.1-c1e2-0-2
Degree $4$
Conductor $1142784$
Sign $1$
Analytic cond. $72.8648$
Root an. cond. $2.92165$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·7-s − 2·9-s + 5·13-s + 7·19-s + 3·21-s − 7·25-s − 5·27-s + 4·31-s + 9·37-s + 5·39-s + 2·43-s − 7·49-s + 7·57-s + 11·61-s − 6·63-s − 10·67-s + 13·73-s − 7·75-s − 9·79-s + 81-s + 15·91-s + 4·93-s − 2·97-s − 25·103-s + 9·109-s + 9·111-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s − 2/3·9-s + 1.38·13-s + 1.60·19-s + 0.654·21-s − 7/5·25-s − 0.962·27-s + 0.718·31-s + 1.47·37-s + 0.800·39-s + 0.304·43-s − 49-s + 0.927·57-s + 1.40·61-s − 0.755·63-s − 1.22·67-s + 1.52·73-s − 0.808·75-s − 1.01·79-s + 1/9·81-s + 1.57·91-s + 0.414·93-s − 0.203·97-s − 2.46·103-s + 0.862·109-s + 0.854·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1142784\)    =    \(2^{12} \cdot 3^{2} \cdot 31\)
Sign: $1$
Analytic conductor: \(72.8648\)
Root analytic conductor: \(2.92165\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1142784} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1142784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.245342418\)
\(L(\frac12)\) \(\approx\) \(3.245342418\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 5 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \)
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 157 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.071961672527906863140337165001, −7.88397742060837631683710051396, −7.27708068995388789561234657196, −6.84968875481701167403000759646, −6.11122279755619925536658942099, −5.81311481108806217063450475162, −5.52850166800920951682541086700, −4.84574940770026474475340192315, −4.40660693366901712230766505898, −3.84015841320905354650239051908, −3.33489208676285362956347948895, −2.89020100760438059253021076952, −2.15312807742855363333262914237, −1.56284056022107850365386695135, −0.836757933859795967155902860058, 0.836757933859795967155902860058, 1.56284056022107850365386695135, 2.15312807742855363333262914237, 2.89020100760438059253021076952, 3.33489208676285362956347948895, 3.84015841320905354650239051908, 4.40660693366901712230766505898, 4.84574940770026474475340192315, 5.52850166800920951682541086700, 5.81311481108806217063450475162, 6.11122279755619925536658942099, 6.84968875481701167403000759646, 7.27708068995388789561234657196, 7.88397742060837631683710051396, 8.071961672527906863140337165001

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