Properties

Label 4-192e2-1.1-c8e2-0-7
Degree $4$
Conductor $36864$
Sign $1$
Analytic cond. $6117.85$
Root an. cond. $8.84402$
Motivic weight $8$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 126·3-s + 5.57e3·7-s + 9.31e3·9-s + 2.63e4·13-s − 2.88e5·19-s + 7.02e5·21-s + 4.48e5·25-s + 3.47e5·27-s + 1.45e6·31-s + 3.92e6·37-s + 3.31e6·39-s + 1.56e5·43-s + 1.17e7·49-s − 3.62e7·57-s − 3.51e7·61-s + 5.19e7·63-s + 3.42e7·67-s + 5.62e7·73-s + 5.64e7·75-s + 1.83e7·79-s − 1.73e7·81-s + 1.46e8·91-s + 1.83e8·93-s − 2.57e8·97-s − 5.43e8·109-s + 4.95e8·111-s + 2.44e8·117-s + ⋯
L(s)  = 1  + 14/9·3-s + 2.32·7-s + 1.41·9-s + 0.920·13-s − 2.20·19-s + 3.60·21-s + 1.14·25-s + 0.652·27-s + 1.57·31-s + 2.09·37-s + 1.43·39-s + 0.0457·43-s + 2.03·49-s − 3.43·57-s − 2.53·61-s + 3.29·63-s + 1.70·67-s + 1.98·73-s + 1.78·75-s + 0.471·79-s − 0.404·81-s + 2.13·91-s + 2.45·93-s − 2.90·97-s − 3.85·109-s + 3.26·111-s + 1.30·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36864\)    =    \(2^{12} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(6117.85\)
Root analytic conductor: \(8.84402\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36864,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(10.53230031\)
\(L(\frac12)\) \(\approx\) \(10.53230031\)
\(L(5)\) 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$C_2$ \( 1 - 14 p^{2} T + p^{8} T^{2} \)
good5$C_2^2$ \( 1 - 448322 T^{2} + p^{16} T^{4} \)
7$C_2$ \( ( 1 - 398 p T + p^{8} T^{2} )^{2} \)
11$C_2^2$ \( 1 + 74615230 T^{2} + p^{16} T^{4} \)
13$C_2$ \( ( 1 - 13150 T + p^{8} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 9544036610 T^{2} + p^{16} T^{4} \)
19$C_2$ \( ( 1 + 144002 T + p^{8} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 154186508930 T^{2} + p^{16} T^{4} \)
29$C_2^2$ \( 1 - 606859926722 T^{2} + p^{16} T^{4} \)
31$C_2$ \( ( 1 - 728738 T + p^{8} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 1964446 T + p^{8} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 14997407035010 T^{2} + p^{16} T^{4} \)
43$C_2$ \( ( 1 - 78142 T + p^{8} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 35234322443522 T^{2} + p^{16} T^{4} \)
53$C_2^2$ \( 1 - 124246846237250 T^{2} + p^{16} T^{4} \)
59$C_2^2$ \( 1 - 268618401162050 T^{2} + p^{16} T^{4} \)
61$C_2$ \( ( 1 + 17578274 T + p^{8} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 17136766 T + p^{8} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 621182343784322 T^{2} + p^{16} T^{4} \)
73$C_2$ \( ( 1 - 28139330 T + p^{8} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 9182498 T + p^{8} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 3083295701563966 T^{2} + p^{16} T^{4} \)
89$C_2^2$ \( 1 - 1271157775602050 T^{2} + p^{16} T^{4} \)
97$C_2$ \( ( 1 + 128722558 T + p^{8} T^{2} )^{2} \)
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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

−11.07415802538467685725786967020, −10.87060561907954219337643829243, −10.51878753865098720396752250156, −9.554572627390398907535829564576, −9.261478414175280881232027629287, −8.449226092047436527079356756503, −8.334194332319305739279563924720, −8.057646691054584275283450093147, −7.63480016995175457090209532669, −6.61095277823439394937090352960, −6.40481393723842919864285381344, −5.36719108967065396054304662377, −4.76477987223006973184511797753, −4.16708175941653823916283779807, −4.04333501606277030658064774272, −2.80747547042425848016372880626, −2.56304117078828534795578173810, −1.72327712118519718279911833189, −1.43889754811256599737355770903, −0.67813991722413538458649308504, 0.67813991722413538458649308504, 1.43889754811256599737355770903, 1.72327712118519718279911833189, 2.56304117078828534795578173810, 2.80747547042425848016372880626, 4.04333501606277030658064774272, 4.16708175941653823916283779807, 4.76477987223006973184511797753, 5.36719108967065396054304662377, 6.40481393723842919864285381344, 6.61095277823439394937090352960, 7.63480016995175457090209532669, 8.057646691054584275283450093147, 8.334194332319305739279563924720, 8.449226092047436527079356756503, 9.261478414175280881232027629287, 9.554572627390398907535829564576, 10.51878753865098720396752250156, 10.87060561907954219337643829243, 11.07415802538467685725786967020

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