Properties

Label 4-75e2-1.1-c4e2-0-2
Degree $4$
Conductor $5625$
Sign $1$
Analytic cond. $60.1050$
Root an. cond. $2.78437$
Motivic weight $4$
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  − 10·3-s + 18·4-s − 150·7-s + 19·9-s − 180·12-s + 110·13-s + 68·16-s − 694·19-s + 1.50e3·21-s + 620·27-s − 2.70e3·28-s − 6·31-s + 342·36-s − 4.46e3·37-s − 1.10e3·39-s − 2.95e3·43-s − 680·48-s + 1.20e4·49-s + 1.98e3·52-s + 6.94e3·57-s + 734·61-s − 2.85e3·63-s − 3.38e3·64-s − 4.47e3·67-s + 1.39e4·73-s − 1.24e4·76-s + 9.03e3·79-s + ⋯
L(s)  = 1  − 1.11·3-s + 9/8·4-s − 3.06·7-s + 0.234·9-s − 5/4·12-s + 0.650·13-s + 0.265·16-s − 1.92·19-s + 3.40·21-s + 0.850·27-s − 3.44·28-s − 0.00624·31-s + 0.263·36-s − 3.25·37-s − 0.723·39-s − 1.59·43-s − 0.295·48-s + 5.02·49-s + 0.732·52-s + 2.13·57-s + 0.197·61-s − 0.718·63-s − 0.826·64-s − 0.995·67-s + 2.61·73-s − 2.16·76-s + 1.44·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(60.1050\)
Root analytic conductor: \(2.78437\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5625,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3251747785\)
\(L(\frac12)\) \(\approx\) \(0.3251747785\)
\(L(3)\) 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)$
bad3$C_2$ \( 1 + 10 T + p^{4} T^{2} \)
5 \( 1 \)
good2$C_2^2$ \( 1 - 9 p T^{2} + p^{8} T^{4} \)
7$C_2$ \( ( 1 + 75 T + p^{4} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 27882 T^{2} + p^{8} T^{4} \)
13$C_2$ \( ( 1 - 55 T + p^{4} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 84342 T^{2} + p^{8} T^{4} \)
19$C_2$ \( ( 1 + 347 T + p^{4} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 135818 T^{2} + p^{8} T^{4} \)
29$C_2^2$ \( 1 - 673962 T^{2} + p^{8} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p^{4} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2230 T + p^{4} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 778122 T^{2} + p^{8} T^{4} \)
43$C_2$ \( ( 1 + 1475 T + p^{4} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6315138 T^{2} + p^{8} T^{4} \)
53$C_2^2$ \( 1 - 15482538 T^{2} + p^{8} T^{4} \)
59$C_2^2$ \( 1 - 16148322 T^{2} + p^{8} T^{4} \)
61$C_2$ \( ( 1 - 367 T + p^{4} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2235 T + p^{4} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 50586762 T^{2} + p^{8} T^{4} \)
73$C_2$ \( ( 1 - 6970 T + p^{4} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4518 T + p^{4} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 94817858 T^{2} + p^{8} T^{4} \)
89$C_2^2$ \( 1 - 60166082 T^{2} + p^{8} T^{4} \)
97$C_2$ \( ( 1 - 4535 T + p^{4} T^{2} )^{2} \)
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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

−14.93320053043451369821846427837, −13.63787329863982183911772666622, −12.71444910413686537396472802178, −12.24670424924343081835204283642, −12.23077006619931788107679070760, −11.14569491001750393313010402913, −10.83433226615739614266427360774, −10.20297185979772001219747844934, −9.911746794208443932010240305037, −8.942210756351437794241868987255, −8.527419689682406744321453917805, −7.11344847647899500042145943840, −6.75198552205730016386270607384, −6.26375134328120116119979144219, −6.13279780986547954202361990628, −5.06352790818194116330058636110, −3.67011260935714900727613187365, −3.21035613903910152399906075516, −2.10711566850081869116974467627, −0.29058888730628904624096156182, 0.29058888730628904624096156182, 2.10711566850081869116974467627, 3.21035613903910152399906075516, 3.67011260935714900727613187365, 5.06352790818194116330058636110, 6.13279780986547954202361990628, 6.26375134328120116119979144219, 6.75198552205730016386270607384, 7.11344847647899500042145943840, 8.527419689682406744321453917805, 8.942210756351437794241868987255, 9.911746794208443932010240305037, 10.20297185979772001219747844934, 10.83433226615739614266427360774, 11.14569491001750393313010402913, 12.23077006619931788107679070760, 12.24670424924343081835204283642, 12.71444910413686537396472802178, 13.63787329863982183911772666622, 14.93320053043451369821846427837

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