Properties

Label 8-273e4-1.1-c2e4-0-5
Degree $8$
Conductor $5554571841$
Sign $1$
Analytic cond. $3061.89$
Root an. cond. $2.72740$
Motivic weight $2$
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  + 9·9-s + 16·16-s + 52·19-s + 118·31-s − 94·37-s − 71·49-s + 218·67-s + 92·73-s − 4·97-s − 314·103-s + 286·109-s + 127-s + 131-s + 137-s + 139-s + 144·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s + 468·171-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 9-s + 16-s + 2.73·19-s + 3.80·31-s − 2.54·37-s − 1.44·49-s + 3.25·67-s + 1.26·73-s − 0.0412·97-s − 3.04·103-s + 2.62·109-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.13·169-s + 2.73·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3061.89\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.262532135\)
\(L(\frac12)\) \(\approx\) \(5.262532135\)
\(L(2)\) 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)$
bad3$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
7$C_2^2$ \( 1 + 71 T^{2} + p^{4} T^{4} \)
13$C_2^2$ \( 1 + 191 T^{2} + p^{4} T^{4} \)
good2$C_2^3$ \( 1 - p^{4} T^{4} + p^{8} T^{8} \)
5$C_2^3$ \( 1 - p^{4} T^{4} + p^{8} T^{8} \)
11$C_2^3$ \( 1 - p^{4} T^{4} + p^{8} T^{8} \)
17$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
19$C_2$$\times$$C_2^2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2}( 1 + 647 T^{2} + p^{4} T^{4} ) \)
23$C_2^2$ \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
31$C_2$$\times$$C_2^2$ \( ( 1 - 59 T + p^{2} T^{2} )^{2}( 1 + 194 T^{2} + p^{4} T^{4} ) \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 47 T + p^{2} T^{2} )^{2}( 1 - 2062 T^{2} + p^{4} T^{4} ) \)
41$C_2^2$ \( ( 1 + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 61 T + p^{2} T^{2} )^{2}( 1 + 61 T + p^{2} T^{2} )^{2} \)
47$C_2^3$ \( 1 - p^{4} T^{4} + p^{8} T^{8} \)
53$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
59$C_2^3$ \( 1 - p^{4} T^{4} + p^{8} T^{8} \)
61$C_2^2$$\times$$C_2^2$ \( ( 1 - 5233 T^{2} + p^{4} T^{4} )( 1 + 7199 T^{2} + p^{4} T^{4} ) \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 109 T + p^{2} T^{2} )^{2}( 1 + 5906 T^{2} + p^{4} T^{4} ) \)
71$C_2^2$ \( ( 1 + p^{4} T^{4} )^{2} \)
73$C_2$$\times$$C_2^2$ \( ( 1 - 46 T + p^{2} T^{2} )^{2}( 1 + 9791 T^{2} + p^{4} T^{4} ) \)
79$C_2^2$$\times$$C_2^2$ \( ( 1 + 4679 T^{2} + p^{4} T^{4} )( 1 + 7682 T^{2} + p^{4} T^{4} ) \)
83$C_2^2$ \( ( 1 + p^{4} T^{4} )^{2} \)
89$C_2^3$ \( 1 - p^{4} T^{4} + p^{8} T^{8} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2}( 1 - 18814 T^{2} + p^{4} T^{4} ) \)
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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

−8.384765814383948089840903854897, −8.247058705172020003611239709866, −7.911137867410253383857419764743, −7.77991353546681951134629581472, −7.33029799332427413879615498898, −7.09752602356453357821144918682, −6.89776283371867428835503160521, −6.67095501971478642541053094399, −6.38261941202552823616381130482, −6.09925877321317708281355820414, −5.63639113916570491651598829801, −5.43007162644116976317835418531, −5.11486201432553822497255617137, −4.89790164824415717264062031308, −4.73668028424803800956613174604, −4.12852855208055024909064656939, −3.96975280711122456357012974604, −3.40847788399291534029666496749, −3.33356049330524813834588894221, −2.88830161171803259983291766058, −2.62321282634927445517971744446, −1.83579988855513057282352464341, −1.58282220319557095390234888937, −0.865891278954084216486077151237, −0.78665912017727081419356934983, 0.78665912017727081419356934983, 0.865891278954084216486077151237, 1.58282220319557095390234888937, 1.83579988855513057282352464341, 2.62321282634927445517971744446, 2.88830161171803259983291766058, 3.33356049330524813834588894221, 3.40847788399291534029666496749, 3.96975280711122456357012974604, 4.12852855208055024909064656939, 4.73668028424803800956613174604, 4.89790164824415717264062031308, 5.11486201432553822497255617137, 5.43007162644116976317835418531, 5.63639113916570491651598829801, 6.09925877321317708281355820414, 6.38261941202552823616381130482, 6.67095501971478642541053094399, 6.89776283371867428835503160521, 7.09752602356453357821144918682, 7.33029799332427413879615498898, 7.77991353546681951134629581472, 7.911137867410253383857419764743, 8.247058705172020003611239709866, 8.384765814383948089840903854897

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