Properties

Label 4-259200-1.1-c1e2-0-48
Degree $4$
Conductor $259200$
Sign $1$
Analytic cond. $16.5268$
Root an. cond. $2.01626$
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  + 4·11-s + 8·13-s + 16·23-s + 25-s + 16·47-s − 10·49-s − 28·59-s − 28·61-s + 24·71-s + 12·73-s + 8·83-s − 28·97-s − 24·107-s + 4·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯
L(s)  = 1  + 1.20·11-s + 2.21·13-s + 3.33·23-s + 1/5·25-s + 2.33·47-s − 1.42·49-s − 3.64·59-s − 3.58·61-s + 2.84·71-s + 1.40·73-s + 0.878·83-s − 2.84·97-s − 2.32·107-s + 0.383·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-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 + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(259200\)    =    \(2^{7} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(16.5268\)
Root analytic conductor: \(2.01626\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{259200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 259200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.375021220\)
\(L(\frac12)\) \(\approx\) \(2.375021220\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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

−8.968329397033630238742013400050, −8.687154242699380983384182548901, −7.84971135862379121268691116066, −7.70282942257969207736023389997, −6.75835720857323144112923149074, −6.58174289188668174000367417422, −6.24245170247766382943086856462, −5.52198466275353226488973122049, −4.99805546290704735001076450321, −4.40097747366580162117975084188, −3.80379273689590657983398796576, −3.26490524893388333289715416456, −2.79232546255480664151890774513, −1.39528505582892974646328461150, −1.18512273608839156013487668024, 1.18512273608839156013487668024, 1.39528505582892974646328461150, 2.79232546255480664151890774513, 3.26490524893388333289715416456, 3.80379273689590657983398796576, 4.40097747366580162117975084188, 4.99805546290704735001076450321, 5.52198466275353226488973122049, 6.24245170247766382943086856462, 6.58174289188668174000367417422, 6.75835720857323144112923149074, 7.70282942257969207736023389997, 7.84971135862379121268691116066, 8.687154242699380983384182548901, 8.968329397033630238742013400050

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