Properties

Label 4-5292e2-1.1-c1e2-0-13
Degree $4$
Conductor $28005264$
Sign $1$
Analytic cond. $1785.63$
Root an. cond. $6.50052$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·11-s + 3·13-s + 14·17-s − 10·19-s + 4·23-s + 5·25-s − 29-s − 3·31-s + 22·37-s + 9·41-s − 5·43-s − 3·47-s − 6·53-s − 8·55-s + 7·59-s + 3·61-s − 6·65-s − 13·67-s + 16·71-s − 14·73-s + 9·79-s − 83-s − 28·85-s + 30·89-s + 20·95-s − 17·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s + 0.832·13-s + 3.39·17-s − 2.29·19-s + 0.834·23-s + 25-s − 0.185·29-s − 0.538·31-s + 3.61·37-s + 1.40·41-s − 0.762·43-s − 0.437·47-s − 0.824·53-s − 1.07·55-s + 0.911·59-s + 0.384·61-s − 0.744·65-s − 1.58·67-s + 1.89·71-s − 1.63·73-s + 1.01·79-s − 0.109·83-s − 3.03·85-s + 3.17·89-s + 2.05·95-s − 1.72·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28005264\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1785.63\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28005264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.933243138\)
\(L(\frac12)\) \(\approx\) \(3.933243138\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−8.187828525982981068499444327334, −7.931676395544953236966674735474, −7.78526603135592780587181944990, −7.45060997636354154487219080316, −6.97903817468824076086733755140, −6.41719282161575892942509193761, −6.17598240454344442940570901985, −6.11962553044319164611175347136, −5.44851101758007809096921599802, −5.15809943197861207258692524444, −4.47513793999166938105946293531, −4.35102022701928291115922571930, −3.74681453293789987868940252277, −3.71593576267856666396638783750, −3.02783595385104151264205309093, −2.87851373190549796172974051975, −2.07020576387531820079886064849, −1.50200045185499493019281639102, −0.858519095714874694257790457167, −0.75528049050771825133950652097, 0.75528049050771825133950652097, 0.858519095714874694257790457167, 1.50200045185499493019281639102, 2.07020576387531820079886064849, 2.87851373190549796172974051975, 3.02783595385104151264205309093, 3.71593576267856666396638783750, 3.74681453293789987868940252277, 4.35102022701928291115922571930, 4.47513793999166938105946293531, 5.15809943197861207258692524444, 5.44851101758007809096921599802, 6.11962553044319164611175347136, 6.17598240454344442940570901985, 6.41719282161575892942509193761, 6.97903817468824076086733755140, 7.45060997636354154487219080316, 7.78526603135592780587181944990, 7.931676395544953236966674735474, 8.187828525982981068499444327334

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