Properties

Label 4-12e6-1.1-c1e2-0-11
Degree $4$
Conductor $2985984$
Sign $1$
Analytic cond. $190.388$
Root an. cond. $3.71458$
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  + 10·13-s + 10·25-s − 22·37-s − 13·49-s + 2·61-s + 14·73-s + 38·97-s + 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2.77·13-s + 2·25-s − 3.61·37-s − 1.85·49-s + 0.256·61-s + 1.63·73-s + 3.85·97-s + 0.383·109-s − 2·121-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 + 3.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(190.388\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2985984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.702876088\)
\(L(\frac12)\) \(\approx\) \(2.702876088\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )^{2} \)
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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

−9.179117645641228085374461407739, −9.112223179559070144782893456098, −8.761492191281057470668963627567, −8.311768040010242561740227027625, −8.206281514278240616285755181634, −7.56089527598095856668717856638, −6.93870982405224431352355528219, −6.77023319392183657572526330163, −6.30209165720759638904061657559, −6.04119973376921399667922473197, −5.42498258955546119253794807754, −4.90170804215799989605392620031, −4.81737287889976349914890574103, −3.82844166399980370741968220918, −3.55866893260058756196291698298, −3.37196803332135349648728696540, −2.65376227532883829966412341319, −1.74627801398433593820702445156, −1.44693117457659556241888629086, −0.66548116290545091716965141568, 0.66548116290545091716965141568, 1.44693117457659556241888629086, 1.74627801398433593820702445156, 2.65376227532883829966412341319, 3.37196803332135349648728696540, 3.55866893260058756196291698298, 3.82844166399980370741968220918, 4.81737287889976349914890574103, 4.90170804215799989605392620031, 5.42498258955546119253794807754, 6.04119973376921399667922473197, 6.30209165720759638904061657559, 6.77023319392183657572526330163, 6.93870982405224431352355528219, 7.56089527598095856668717856638, 8.206281514278240616285755181634, 8.311768040010242561740227027625, 8.761492191281057470668963627567, 9.112223179559070144782893456098, 9.179117645641228085374461407739

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