Properties

Degree $4$
Conductor $129600$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 4·11-s − 16·19-s + 11·25-s + 16·29-s − 4·41-s + 10·49-s + 16·55-s − 12·59-s + 4·61-s + 8·71-s + 16·79-s + 12·89-s + 64·95-s + 12·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.20·11-s − 3.67·19-s + 11/5·25-s + 2.97·29-s − 0.624·41-s + 10/7·49-s + 2.15·55-s − 1.56·59-s + 0.512·61-s + 0.949·71-s + 1.80·79-s + 1.27·89-s + 6.56·95-s + 1.14·109-s − 0.909·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 129600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620260\)
\(L(\frac12)\) \(\approx\) \(0.620260\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−11.80210888241706262621298027878, −10.97047280000368757684386671512, −10.80040561276049023925394338788, −10.49328321214195567745791798463, −10.13132762261123793513291037470, −9.200996140699861748593738318899, −8.585616972306893103939095061379, −8.281361488778413926463002734989, −8.217447225720674731267822909093, −7.51141579759640834159242484090, −6.93043854564591477779557992318, −6.45518930545176223838856956012, −6.08256025102826789712467462417, −4.94091759316762413426731784931, −4.70015757120587292748410980883, −4.19791095133386478732424228648, −3.60070950372770491348534872187, −2.77212848526543163278910110492, −2.21230070927819415853594571671, −0.52628682333098433130296392786, 0.52628682333098433130296392786, 2.21230070927819415853594571671, 2.77212848526543163278910110492, 3.60070950372770491348534872187, 4.19791095133386478732424228648, 4.70015757120587292748410980883, 4.94091759316762413426731784931, 6.08256025102826789712467462417, 6.45518930545176223838856956012, 6.93043854564591477779557992318, 7.51141579759640834159242484090, 8.217447225720674731267822909093, 8.281361488778413926463002734989, 8.585616972306893103939095061379, 9.200996140699861748593738318899, 10.13132762261123793513291037470, 10.49328321214195567745791798463, 10.80040561276049023925394338788, 10.97047280000368757684386671512, 11.80210888241706262621298027878

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