Properties

Label 4-180e2-1.1-c1e2-0-6
Degree $4$
Conductor $32400$
Sign $1$
Analytic cond. $2.06585$
Root an. cond. $1.19887$
Motivic weight $1$
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  + 3·3-s − 5-s + 7-s + 6·9-s + 4·13-s − 3·15-s − 12·17-s + 4·19-s + 3·21-s + 3·23-s + 9·27-s − 3·29-s + 10·31-s − 35-s − 20·37-s + 12·39-s − 9·41-s + 4·43-s − 6·45-s − 9·47-s + 7·49-s − 36·51-s − 12·53-s + 12·57-s + 6·59-s + 61-s + 6·63-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.10·13-s − 0.774·15-s − 2.91·17-s + 0.917·19-s + 0.654·21-s + 0.625·23-s + 1.73·27-s − 0.557·29-s + 1.79·31-s − 0.169·35-s − 3.28·37-s + 1.92·39-s − 1.40·41-s + 0.609·43-s − 0.894·45-s − 1.31·47-s + 49-s − 5.04·51-s − 1.64·53-s + 1.58·57-s + 0.781·59-s + 0.128·61-s + 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.06585\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 32400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.145200731\)
\(L(\frac12)\) \(\approx\) \(2.145200731\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−13.07323100536743089832902151211, −12.64695803371730144828687322633, −11.72487625931839831993825130033, −11.61554228388570923308019590650, −10.71872216541710044944359507107, −10.59624379239894998133655309897, −9.764251191120569755496157805707, −9.131490110610312350342747870835, −8.669759085305776939208420995501, −8.637834542706514113266809170384, −7.923449841738607293638115484093, −7.39525228783872442131807080311, −6.57801519763663424795933877212, −6.54779774330716892423916671853, −5.02867996757359366970880323523, −4.71978973954967404955287113488, −3.72024618991411865920399568644, −3.46873584516959243507074780252, −2.43901192103932383202408943676, −1.67886815582823748703660128397, 1.67886815582823748703660128397, 2.43901192103932383202408943676, 3.46873584516959243507074780252, 3.72024618991411865920399568644, 4.71978973954967404955287113488, 5.02867996757359366970880323523, 6.54779774330716892423916671853, 6.57801519763663424795933877212, 7.39525228783872442131807080311, 7.923449841738607293638115484093, 8.637834542706514113266809170384, 8.669759085305776939208420995501, 9.131490110610312350342747870835, 9.764251191120569755496157805707, 10.59624379239894998133655309897, 10.71872216541710044944359507107, 11.61554228388570923308019590650, 11.72487625931839831993825130033, 12.64695803371730144828687322633, 13.07323100536743089832902151211

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