Properties

Label 4-20e4-1.1-c3e2-0-13
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $556.996$
Root an. cond. $4.85806$
Motivic weight $3$
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  + 53·9-s + 38·11-s − 182·19-s + 544·29-s + 460·31-s + 234·41-s + 650·49-s + 624·59-s + 340·61-s + 104·71-s + 2.10e3·79-s + 2.08e3·81-s − 1.59e3·89-s + 2.01e3·99-s + 972·101-s − 252·109-s − 1.57e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.25e3·169-s + ⋯
L(s)  = 1  + 1.96·9-s + 1.04·11-s − 2.19·19-s + 3.48·29-s + 2.66·31-s + 0.891·41-s + 1.89·49-s + 1.37·59-s + 0.713·61-s + 0.173·71-s + 3.00·79-s + 2.85·81-s − 1.90·89-s + 2.04·99-s + 0.957·101-s − 0.221·109-s − 1.18·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.93·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(556.996\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.401578513\)
\(L(\frac12)\) \(\approx\) \(4.401578513\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 53 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 650 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 - 19 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4250 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 4201 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 91 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 5942 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 272 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 230 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 68182 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 117 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 20630 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 204942 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 136150 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 312 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 170 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 19357 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 52 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 184327 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 1054 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 1020373 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 799 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 899902 T^{2} + p^{6} T^{4} \)
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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

−10.88559011104797266410111678471, −10.43216323156229785563894629932, −10.24114590120319901937816795921, −9.912968364220503359935906572824, −9.253911627954725662953415832923, −8.793472451952456622956243901090, −8.162322034939948061963859831206, −8.121759004001235444547007582332, −7.13871491721883781920135605280, −6.77005320420479934354302693853, −6.37764305951255693022615159357, −6.21947650904631683446324341455, −4.95996279216386443526963628436, −4.66331691881455364681238946425, −4.02999826829729584887079666346, −3.95612282990495549841317380376, −2.63969992106972965485000595024, −2.27698416094884778906070607609, −1.09704500736939351538130665997, −0.918650137242675832922380449169, 0.918650137242675832922380449169, 1.09704500736939351538130665997, 2.27698416094884778906070607609, 2.63969992106972965485000595024, 3.95612282990495549841317380376, 4.02999826829729584887079666346, 4.66331691881455364681238946425, 4.95996279216386443526963628436, 6.21947650904631683446324341455, 6.37764305951255693022615159357, 6.77005320420479934354302693853, 7.13871491721883781920135605280, 8.121759004001235444547007582332, 8.162322034939948061963859831206, 8.793472451952456622956243901090, 9.253911627954725662953415832923, 9.912968364220503359935906572824, 10.24114590120319901937816795921, 10.43216323156229785563894629932, 10.88559011104797266410111678471

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