Properties

Label 4-360e2-1.1-c1e2-0-32
Degree $4$
Conductor $129600$
Sign $1$
Analytic cond. $8.26340$
Root an. cond. $1.69546$
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  + 4·7-s + 8·13-s + 8·19-s + 25-s + 8·31-s − 16·43-s − 2·49-s − 28·61-s − 8·67-s + 12·73-s − 24·79-s + 32·91-s − 28·97-s − 28·103-s + 4·109-s − 18·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 1.51·7-s + 2.21·13-s + 1.83·19-s + 1/5·25-s + 1.43·31-s − 2.43·43-s − 2/7·49-s − 3.58·61-s − 0.977·67-s + 1.40·73-s − 2.70·79-s + 3.35·91-s − 2.84·97-s − 2.75·103-s + 0.383·109-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(129600\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(8.26340\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{129600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 129600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.250175564\)
\(L(\frac12)\) \(\approx\) \(2.250175564\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−9.419368133385122785233890515409, −8.687154242699380983384182548901, −8.430531363871330384753929024064, −7.942341734598117376456164094334, −7.70282942257969207736023389997, −6.75835720857323144112923149074, −6.53876572552795316440603698045, −5.59184082567252664246215870943, −5.52198466275353226488973122049, −4.54321125337695258384863067085, −4.40097747366580162117975084188, −3.26490524893388333289715416456, −3.08000224242427569012154075457, −1.50557401701584832617156823174, −1.39528505582892974646328461150, 1.39528505582892974646328461150, 1.50557401701584832617156823174, 3.08000224242427569012154075457, 3.26490524893388333289715416456, 4.40097747366580162117975084188, 4.54321125337695258384863067085, 5.52198466275353226488973122049, 5.59184082567252664246215870943, 6.53876572552795316440603698045, 6.75835720857323144112923149074, 7.70282942257969207736023389997, 7.942341734598117376456164094334, 8.430531363871330384753929024064, 8.687154242699380983384182548901, 9.419368133385122785233890515409

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