Properties

Label 4-1036800-1.1-c1e2-0-15
Degree $4$
Conductor $1036800$
Sign $-1$
Analytic cond. $66.1072$
Root an. cond. $2.85142$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s − 4·17-s + 8·19-s + 25-s − 16·43-s − 10·49-s − 28·59-s − 8·67-s + 12·73-s + 8·83-s − 24·89-s − 28·97-s − 24·107-s + 36·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.970·17-s + 1.83·19-s + 1/5·25-s − 2.43·43-s − 1.42·49-s − 3.64·59-s − 0.977·67-s + 1.40·73-s + 0.878·83-s − 2.54·89-s − 2.84·97-s − 2.32·107-s + 3.38·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-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 − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1036800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1036800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1036800\)    =    \(2^{9} \cdot 3^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(66.1072\)
Root analytic conductor: \(2.85142\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1036800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1036800,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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

−7.70282942257969207736023389997, −7.63755234700784730433588895727, −6.75835720857323144112923149074, −6.67196691618067429234936685096, −6.31926233328074557516075297223, −5.52198466275353226488973122049, −5.28153821016403058290609790842, −4.52573962616420476871662034072, −4.40097747366580162117975084188, −3.48586526401719520000666177669, −3.26490524893388333289715416456, −2.63989707037125425260719850316, −1.57588123883597793160522483813, −1.39528505582892974646328461150, 0, 1.39528505582892974646328461150, 1.57588123883597793160522483813, 2.63989707037125425260719850316, 3.26490524893388333289715416456, 3.48586526401719520000666177669, 4.40097747366580162117975084188, 4.52573962616420476871662034072, 5.28153821016403058290609790842, 5.52198466275353226488973122049, 6.31926233328074557516075297223, 6.67196691618067429234936685096, 6.75835720857323144112923149074, 7.63755234700784730433588895727, 7.70282942257969207736023389997

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