Properties

Label 4-4140e2-1.1-c1e2-0-2
Degree $4$
Conductor $17139600$
Sign $1$
Analytic cond. $1092.83$
Root an. cond. $5.74961$
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 + 2·7-s + 4·11-s + 2·17-s − 2·23-s + 3·25-s + 14·29-s + 6·31-s + 4·35-s − 10·37-s + 6·41-s + 4·43-s − 11·49-s + 6·53-s + 8·55-s + 18·59-s − 4·61-s − 2·67-s + 10·71-s + 8·77-s − 8·79-s + 26·83-s + 4·85-s + 16·89-s + 18·101-s − 4·103-s − 18·107-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 1.20·11-s + 0.485·17-s − 0.417·23-s + 3/5·25-s + 2.59·29-s + 1.07·31-s + 0.676·35-s − 1.64·37-s + 0.937·41-s + 0.609·43-s − 1.57·49-s + 0.824·53-s + 1.07·55-s + 2.34·59-s − 0.512·61-s − 0.244·67-s + 1.18·71-s + 0.911·77-s − 0.900·79-s + 2.85·83-s + 0.433·85-s + 1.69·89-s + 1.79·101-s − 0.394·103-s − 1.74·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17139600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1092.83\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17139600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.518733238\)
\(L(\frac12)\) \(\approx\) \(5.518733238\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 14 T + 101 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 61 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 18 T + 193 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 113 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 26 T + 329 T^{2} - 26 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 170 T^{2} + p^{2} T^{4} \)
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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

−8.486051677170449175993952635482, −8.378843067517269923985759060009, −7.80319161465039942699747762390, −7.73306771042633936488369791715, −6.83920302859602897722186756324, −6.82108938464682541674503827495, −6.31615433910063084842179603795, −6.28376803028986488574203348391, −5.45514262545136575658780105709, −5.39450616536368769383489325358, −4.80136844867913926214685672314, −4.59226438787871234996184266399, −4.04682575812786842521816566539, −3.66026892071206253258113392441, −3.13282321998270431888605023045, −2.62722058986211987942536223678, −2.17984878007080163743100052573, −1.69906004859066753445086354474, −1.08233632397486074791530573349, −0.792486365317966618573500168079, 0.792486365317966618573500168079, 1.08233632397486074791530573349, 1.69906004859066753445086354474, 2.17984878007080163743100052573, 2.62722058986211987942536223678, 3.13282321998270431888605023045, 3.66026892071206253258113392441, 4.04682575812786842521816566539, 4.59226438787871234996184266399, 4.80136844867913926214685672314, 5.39450616536368769383489325358, 5.45514262545136575658780105709, 6.28376803028986488574203348391, 6.31615433910063084842179603795, 6.82108938464682541674503827495, 6.83920302859602897722186756324, 7.73306771042633936488369791715, 7.80319161465039942699747762390, 8.378843067517269923985759060009, 8.486051677170449175993952635482

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