Properties

Label 8-432e4-1.1-c1e4-0-2
Degree $8$
Conductor $34828517376$
Sign $1$
Analytic cond. $141.593$
Root an. cond. $1.85729$
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  − 6·5-s − 2·13-s + 11·25-s + 30·29-s − 16·37-s − 18·41-s − 5·49-s + 14·61-s + 12·65-s + 16·73-s − 2·97-s + 18·101-s + 16·109-s + 42·113-s − 5·121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 180·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 27·169-s + ⋯
L(s)  = 1  − 2.68·5-s − 0.554·13-s + 11/5·25-s + 5.57·29-s − 2.63·37-s − 2.81·41-s − 5/7·49-s + 1.79·61-s + 1.48·65-s + 1.87·73-s − 0.203·97-s + 1.79·101-s + 1.53·109-s + 3.95·113-s − 0.454·121-s + 0.536·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 14.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.07·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(141.593\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{432} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9597194648\)
\(L(\frac12)\) \(\approx\) \(0.9597194648\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^3$ \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 + 5 T^{2} - 96 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 19 T^{2} - 168 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 15 T + 104 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^3$ \( 1 + 53 T^{2} + 1848 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 + 77 T^{2} + 4080 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^3$ \( 1 - 67 T^{2} + 2280 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 53 T^{2} - 1680 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
79$C_2^3$ \( 1 - 67 T^{2} - 1752 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^3$ \( 1 - 139 T^{2} + 12432 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 166 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.210957469752214655922877825642, −7.78279468507188299007012152249, −7.71048529863858397461564260172, −7.43720225186532749497206918951, −6.94703234384423004082346582114, −6.91190831286809726056925378783, −6.70689262348291192495058544204, −6.55107730880931695185568681008, −6.23809981420566085323674454440, −5.82548973559547275081009165802, −5.38946605234373196976226843857, −5.04736710289279896835066056162, −4.94293898891516107615446896574, −4.71045816125836541807855511345, −4.40724668051379413981439624924, −4.21242972291899748900694957756, −3.69337743861735498078780936030, −3.58536186051464766832966500005, −3.27034398913898410384560579836, −2.98602229952351743843644484765, −2.74568514064946745485319457832, −1.89811329018923357890472649700, −1.88561405254783143931675338137, −0.78667881984835800281641273800, −0.52363058145015849694816103396, 0.52363058145015849694816103396, 0.78667881984835800281641273800, 1.88561405254783143931675338137, 1.89811329018923357890472649700, 2.74568514064946745485319457832, 2.98602229952351743843644484765, 3.27034398913898410384560579836, 3.58536186051464766832966500005, 3.69337743861735498078780936030, 4.21242972291899748900694957756, 4.40724668051379413981439624924, 4.71045816125836541807855511345, 4.94293898891516107615446896574, 5.04736710289279896835066056162, 5.38946605234373196976226843857, 5.82548973559547275081009165802, 6.23809981420566085323674454440, 6.55107730880931695185568681008, 6.70689262348291192495058544204, 6.91190831286809726056925378783, 6.94703234384423004082346582114, 7.43720225186532749497206918951, 7.71048529863858397461564260172, 7.78279468507188299007012152249, 8.210957469752214655922877825642

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