Properties

Label 4-209952-1.1-c1e2-0-4
Degree $4$
Conductor $209952$
Sign $-1$
Analytic cond. $13.3867$
Root an. cond. $1.91279$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 6·5-s − 8-s + 6·10-s − 2·13-s + 16-s − 6·17-s − 6·20-s + 17·25-s + 2·26-s + 18·29-s − 32-s + 6·34-s − 2·37-s + 6·40-s + 12·41-s + 2·49-s − 17·50-s − 2·52-s − 12·53-s − 18·58-s − 2·61-s + 64-s + 12·65-s − 6·68-s + 22·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 2.68·5-s − 0.353·8-s + 1.89·10-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 1.34·20-s + 17/5·25-s + 0.392·26-s + 3.34·29-s − 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.948·40-s + 1.87·41-s + 2/7·49-s − 2.40·50-s − 0.277·52-s − 1.64·53-s − 2.36·58-s − 0.256·61-s + 1/8·64-s + 1.48·65-s − 0.727·68-s + 2.57·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(209952\)    =    \(2^{5} \cdot 3^{8}\)
Sign: $-1$
Analytic conductor: \(13.3867\)
Root analytic conductor: \(1.91279\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 209952,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3 \( 1 \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.542414927665885183690210552564, −8.278948375699901432058512087738, −8.047221009677602422962699873621, −7.44571478799237129140639120122, −7.16277862972928710155421440227, −6.50974172394367970294419909175, −6.29261335708839240986736404399, −5.10609983558079824652464594116, −4.60529471992634973001671480527, −4.25638665319375785757127060430, −3.68824329939728705086964001391, −2.94651413139897301734292877776, −2.41615112845105918353887046447, −0.935395355311356120178486862505, 0, 0.935395355311356120178486862505, 2.41615112845105918353887046447, 2.94651413139897301734292877776, 3.68824329939728705086964001391, 4.25638665319375785757127060430, 4.60529471992634973001671480527, 5.10609983558079824652464594116, 6.29261335708839240986736404399, 6.50974172394367970294419909175, 7.16277862972928710155421440227, 7.44571478799237129140639120122, 8.047221009677602422962699873621, 8.278948375699901432058512087738, 8.542414927665885183690210552564

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