Properties

Label 4-192e2-1.1-c2e2-0-1
Degree $4$
Conductor $36864$
Sign $1$
Analytic cond. $27.3698$
Root an. cond. $2.28727$
Motivic weight $2$
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  − 12·5-s − 3·9-s + 28·13-s − 12·17-s + 58·25-s − 60·29-s − 52·37-s − 108·41-s + 36·45-s + 50·49-s + 36·53-s + 140·61-s − 336·65-s + 164·73-s + 9·81-s + 144·85-s + 228·89-s + 68·97-s + 36·101-s − 68·109-s − 156·113-s − 84·117-s − 190·121-s + 36·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2.39·5-s − 1/3·9-s + 2.15·13-s − 0.705·17-s + 2.31·25-s − 2.06·29-s − 1.40·37-s − 2.63·41-s + 4/5·45-s + 1.02·49-s + 0.679·53-s + 2.29·61-s − 5.16·65-s + 2.24·73-s + 1/9·81-s + 1.69·85-s + 2.56·89-s + 0.701·97-s + 0.356·101-s − 0.623·109-s − 1.38·113-s − 0.717·117-s − 1.57·121-s + 0.287·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36864\)    =    \(2^{12} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(27.3698\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36864,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7459479299\)
\(L(\frac12)\) \(\approx\) \(0.7459479299\)
\(L(2)\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 50 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 190 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 674 T^{2} + p^{4} T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1490 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 54 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 3266 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 2690 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6530 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 4894 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 3170 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 6674 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 13346 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 114 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )^{2} \)
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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

−12.31913464371876665447767818138, −11.88835318663780390861657362668, −11.61053766197120616313103490582, −11.18129072154113067986754908535, −10.81365970397423608373755426679, −10.36331169183046435625152979529, −9.400841679566366039215493598883, −8.816981222403210212250512039106, −8.377607145457580686486383334929, −8.188910495025699306118599680339, −7.49852036275646657176237460762, −6.95876608073337558873293685233, −6.48320625867505541434603991069, −5.60603373814645247556759848579, −5.02082383370425517343335080136, −4.00561473197111105332721914577, −3.66740012015501819603540992330, −3.47893809978424182834916222429, −1.93539103365095390518858657712, −0.50814040392929698989456080219, 0.50814040392929698989456080219, 1.93539103365095390518858657712, 3.47893809978424182834916222429, 3.66740012015501819603540992330, 4.00561473197111105332721914577, 5.02082383370425517343335080136, 5.60603373814645247556759848579, 6.48320625867505541434603991069, 6.95876608073337558873293685233, 7.49852036275646657176237460762, 8.188910495025699306118599680339, 8.377607145457580686486383334929, 8.816981222403210212250512039106, 9.400841679566366039215493598883, 10.36331169183046435625152979529, 10.81365970397423608373755426679, 11.18129072154113067986754908535, 11.61053766197120616313103490582, 11.88835318663780390861657362668, 12.31913464371876665447767818138

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