Properties

Label 4-42e4-1.1-c1e2-0-8
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $198.404$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 12·13-s + 4·16-s − 8·25-s − 4·37-s − 24·52-s + 24·61-s − 8·64-s − 12·73-s − 36·97-s + 16·100-s − 40·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 4-s + 3.32·13-s + 16-s − 8/5·25-s − 0.657·37-s − 3.32·52-s + 3.07·61-s − 64-s − 1.40·73-s − 3.65·97-s + 8/5·100-s − 3.83·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(198.404\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.842474238\)
\(L(\frac12)\) \(\approx\) \(1.842474238\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 160 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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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

−9.652173553832694733360080910155, −8.873119540056533245525347234187, −8.801394743501445615550369381552, −8.358622308332705733988312252279, −7.937538777983804333700619065598, −7.916842880453020762746270948022, −6.98576482890637229015149049786, −6.66046356916649474593699749730, −6.26884477414532737926062521435, −5.77045337030886139303586056636, −5.37875314578908481358825924015, −5.31136998390671865165742300088, −4.16319772384089492977424346533, −4.03822964536088680837124253099, −3.89160305151847577774218133684, −3.24958931687781157788739063697, −2.71113413995048840570646202076, −1.64625240467082765857055783044, −1.40433213360464605806209832799, −0.56543345258875710026540621223, 0.56543345258875710026540621223, 1.40433213360464605806209832799, 1.64625240467082765857055783044, 2.71113413995048840570646202076, 3.24958931687781157788739063697, 3.89160305151847577774218133684, 4.03822964536088680837124253099, 4.16319772384089492977424346533, 5.31136998390671865165742300088, 5.37875314578908481358825924015, 5.77045337030886139303586056636, 6.26884477414532737926062521435, 6.66046356916649474593699749730, 6.98576482890637229015149049786, 7.916842880453020762746270948022, 7.937538777983804333700619065598, 8.358622308332705733988312252279, 8.801394743501445615550369381552, 8.873119540056533245525347234187, 9.652173553832694733360080910155

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