Properties

Label 4-292032-1.1-c1e2-0-18
Degree $4$
Conductor $292032$
Sign $1$
Analytic cond. $18.6202$
Root an. cond. $2.07728$
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-s + 9-s + 2·13-s − 10·25-s + 27-s + 4·37-s + 2·39-s − 10·49-s + 24·59-s + 4·61-s + 24·71-s + 28·73-s − 10·75-s + 81-s + 24·83-s − 20·97-s − 24·107-s + 28·109-s + 4·111-s + 2·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 10·147-s + 149-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.554·13-s − 2·25-s + 0.192·27-s + 0.657·37-s + 0.320·39-s − 1.42·49-s + 3.12·59-s + 0.512·61-s + 2.84·71-s + 3.27·73-s − 1.15·75-s + 1/9·81-s + 2.63·83-s − 2.03·97-s − 2.32·107-s + 2.68·109-s + 0.379·111-s + 0.184·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.824·147-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(292032\)    =    \(2^{6} \cdot 3^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(18.6202\)
Root analytic conductor: \(2.07728\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 292032,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.186757681\)
\(L(\frac12)\) \(\approx\) \(2.186757681\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−8.735814680045936069504206621070, −8.330201821850417900078578930922, −7.86871640882441783507560416572, −7.76963269770720756784752102354, −6.78013921434981904340142936703, −6.65843891177652373785895186115, −6.05756828657732859884712721948, −5.31787425835571416225740279689, −5.11454570698520102977754794044, −4.13529224817546208100485895471, −3.80067854107620357152276821608, −3.37281563192054858635234459773, −2.33566034645582102228367652701, −2.03881758362108754018941589683, −0.861350648696305200147674327945, 0.861350648696305200147674327945, 2.03881758362108754018941589683, 2.33566034645582102228367652701, 3.37281563192054858635234459773, 3.80067854107620357152276821608, 4.13529224817546208100485895471, 5.11454570698520102977754794044, 5.31787425835571416225740279689, 6.05756828657732859884712721948, 6.65843891177652373785895186115, 6.78013921434981904340142936703, 7.76963269770720756784752102354, 7.86871640882441783507560416572, 8.330201821850417900078578930922, 8.735814680045936069504206621070

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