Properties

Label 4-6e6-1.1-c5e2-0-4
Degree $4$
Conductor $46656$
Sign $1$
Analytic cond. $1200.13$
Root an. cond. $5.88582$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·5-s + 12·7-s + 238·11-s − 472·13-s + 368·17-s − 56·19-s − 1.02e3·23-s − 4.06e3·25-s − 144·29-s − 8.22e3·31-s − 240·35-s − 1.40e4·37-s + 6.69e3·41-s − 1.51e4·43-s − 156·47-s − 3.16e4·49-s + 3.00e3·53-s − 4.76e3·55-s − 2.77e4·59-s − 5.78e4·61-s + 9.44e3·65-s − 7.41e4·67-s − 5.87e4·71-s − 4.96e4·73-s + 2.85e3·77-s − 8.29e4·79-s − 2.55e4·83-s + ⋯
L(s)  = 1  − 0.357·5-s + 0.0925·7-s + 0.593·11-s − 0.774·13-s + 0.308·17-s − 0.0355·19-s − 0.405·23-s − 1.30·25-s − 0.0317·29-s − 1.53·31-s − 0.0331·35-s − 1.69·37-s + 0.622·41-s − 1.24·43-s − 0.0103·47-s − 1.88·49-s + 0.146·53-s − 0.212·55-s − 1.03·59-s − 1.99·61-s + 0.277·65-s − 2.01·67-s − 1.38·71-s − 1.08·73-s + 0.0548·77-s − 1.49·79-s − 0.407·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(46656\)    =    \(2^{6} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(1200.13\)
Root analytic conductor: \(5.88582\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 46656,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good5$D_{4}$ \( 1 + 4 p T + 4469 T^{2} + 4 p^{6} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 - 12 T + 31769 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 238 T + 65399 T^{2} - 238 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 472 T + 316746 T^{2} + 472 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 368 T + 2392034 T^{2} - 368 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 56 T + 2236818 T^{2} + 56 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 1028 T + 11210738 T^{2} + 1028 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 144 T + 27115606 T^{2} + 144 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 8228 T + 59283897 T^{2} + 8228 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 14076 T + 173654894 T^{2} + 14076 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 6696 T + 214924702 T^{2} - 6696 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 15112 T + 233547522 T^{2} + 15112 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 156 T + 438351202 T^{2} + 156 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 3004 T + 829164869 T^{2} - 3004 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 27704 T + 1552385318 T^{2} + 27704 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 57856 T + 2525901402 T^{2} + 57856 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 74168 T + 4013167026 T^{2} + 74168 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 58768 T + 3739461902 T^{2} + 58768 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 49606 T + 1887735819 T^{2} + 49606 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 82984 T + 6683867166 T^{2} + 82984 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 25570 T + 6145459415 T^{2} + 25570 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 21480 T - 378921602 T^{2} + 21480 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 79826 T + 17389451667 T^{2} + 79826 p^{5} T^{3} + p^{10} T^{4} \)
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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

−11.23061522113011600965813502377, −10.76854398851965594687901909997, −10.12217054022633024633343732159, −9.859785451987491762133768840004, −9.051629860661886120738019645443, −8.960843391827883833466552132351, −8.059451989177219277462692452344, −7.71143256348342263748628032674, −7.18646331862992034566609725019, −6.70710408076084630689405631775, −5.81630499823715039874702461507, −5.65160995449861121461662568716, −4.59185836293005279503307892800, −4.38551121077357894686882789492, −3.39628554635388561437526808280, −3.08650040738896053619626261566, −1.78004823853156828648944268606, −1.60588528703203252114880053910, 0, 0, 1.60588528703203252114880053910, 1.78004823853156828648944268606, 3.08650040738896053619626261566, 3.39628554635388561437526808280, 4.38551121077357894686882789492, 4.59185836293005279503307892800, 5.65160995449861121461662568716, 5.81630499823715039874702461507, 6.70710408076084630689405631775, 7.18646331862992034566609725019, 7.71143256348342263748628032674, 8.059451989177219277462692452344, 8.960843391827883833466552132351, 9.051629860661886120738019645443, 9.859785451987491762133768840004, 10.12217054022633024633343732159, 10.76854398851965594687901909997, 11.23061522113011600965813502377

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