Properties

Label 8-4032e4-1.1-c1e4-0-27
Degree $8$
Conductor $2.643\times 10^{14}$
Sign $1$
Analytic cond. $1.07446\times 10^{6}$
Root an. cond. $5.67412$
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  + 4·7-s − 8·17-s + 12·25-s + 16·31-s + 8·41-s + 16·47-s + 10·49-s + 16·71-s + 24·73-s − 16·79-s + 8·89-s + 8·97-s − 16·103-s + 40·113-s − 32·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.94·17-s + 12/5·25-s + 2.87·31-s + 1.24·41-s + 2.33·47-s + 10/7·49-s + 1.89·71-s + 2.80·73-s − 1.80·79-s + 0.847·89-s + 0.812·97-s − 1.57·103-s + 3.76·113-s − 2.93·119-s + 1.09·121-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 − 0.307·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.07446\times 10^{6}\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.27344565\)
\(L(\frac12)\) \(\approx\) \(10.27344565\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
good5$D_4\times C_2$ \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 + 4 T^{2} + 42 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
19$D_4\times C_2$ \( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 84 T^{2} + 3254 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 116 T^{2} + 5910 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
43$D_4\times C_2$ \( 1 - 140 T^{2} + 8406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 84 T^{2} + 8426 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 52 T^{2} + 7530 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.93321327675320842843106661936, −5.82156660950061288971780459035, −5.50055941324693868980580841267, −5.38749594848974859794813310291, −5.00456195242951721897687773167, −4.82523378632418391122968503421, −4.78866649641898419718472458578, −4.74056694499241160284600486528, −4.43111780330977839070401810911, −4.17554785712838267347377461100, −4.03615721171754609322479084351, −3.76685215261090873670267718222, −3.71670805153783638711646739688, −3.11040902021851877959284452861, −3.04979352671262024151969069171, −2.71594448099081447169076654390, −2.54909342886205905060622449962, −2.45366247578074513418746753260, −2.04514786207183482324839957063, −2.00020415806930153266812922547, −1.57477341267438928231811508174, −1.21530835295068399392295546509, −0.851080905418928133302172024834, −0.75230275504825559925292742269, −0.51308146535668103823880411817, 0.51308146535668103823880411817, 0.75230275504825559925292742269, 0.851080905418928133302172024834, 1.21530835295068399392295546509, 1.57477341267438928231811508174, 2.00020415806930153266812922547, 2.04514786207183482324839957063, 2.45366247578074513418746753260, 2.54909342886205905060622449962, 2.71594448099081447169076654390, 3.04979352671262024151969069171, 3.11040902021851877959284452861, 3.71670805153783638711646739688, 3.76685215261090873670267718222, 4.03615721171754609322479084351, 4.17554785712838267347377461100, 4.43111780330977839070401810911, 4.74056694499241160284600486528, 4.78866649641898419718472458578, 4.82523378632418391122968503421, 5.00456195242951721897687773167, 5.38749594848974859794813310291, 5.50055941324693868980580841267, 5.82156660950061288971780459035, 5.93321327675320842843106661936

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