Properties

Label 8-3330e4-1.1-c1e4-0-8
Degree $8$
Conductor $1.230\times 10^{14}$
Sign $1$
Analytic cond. $499902.$
Root an. cond. $5.15656$
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 + 2·7-s − 14·11-s + 3·16-s − 2·25-s − 4·28-s + 2·41-s + 28·44-s − 9·49-s + 22·53-s − 4·64-s + 12·67-s + 12·71-s + 56·73-s − 28·77-s − 24·83-s + 4·100-s + 44·101-s − 36·107-s + 6·112-s + 95·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 4-s + 0.755·7-s − 4.22·11-s + 3/4·16-s − 2/5·25-s − 0.755·28-s + 0.312·41-s + 4.22·44-s − 9/7·49-s + 3.02·53-s − 1/2·64-s + 1.46·67-s + 1.42·71-s + 6.55·73-s − 3.19·77-s − 2.63·83-s + 2/5·100-s + 4.37·101-s − 3.48·107-s + 0.566·112-s + 8.63·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(499902.\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.244205884\)
\(L(\frac12)\) \(\approx\) \(2.244205884\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3 \( 1 \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
good7$D_{4}$ \( ( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 16 T^{2} + 270 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 39 T^{2} + 752 T^{4} - 39 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 8 T^{2} - 114 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 15 T^{2} + 344 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 47 T^{2} + 1476 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 151 T^{2} + 9324 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$D_{4}$ \( ( 1 - 11 T + 128 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 83 T^{2} + 6780 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 6 T + 118 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
79$D_4\times C_2$ \( 1 - 248 T^{2} + 27726 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 272 T^{2} + 33150 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 235 T^{2} + 31956 T^{4} - 235 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.19445986200286747151343824184, −5.62604856124310365688036902192, −5.57895694820641188693109378265, −5.28093831186049281908105943196, −5.20717155187260899914241134519, −5.18182255883410808915741099613, −5.09977543233331387846820203424, −4.83826683885073884470707487387, −4.62736487492871232532312959884, −4.25637865156061974468331781665, −3.91649540821464401928233952970, −3.86699181156119072108522408239, −3.81876954770496368967671421011, −3.32203759262580233174092837570, −3.21597190498816595202878816593, −2.76833139466050482736431549082, −2.65943536581432967175814939287, −2.28976696499242668668994722569, −2.27124050842289513103068895603, −2.23945790092984610411674453970, −1.50754710852943044894232065404, −1.45960223867565822030913430064, −0.67976050878206954495565325186, −0.53052538117526306189926779702, −0.42015787911658238281104103298, 0.42015787911658238281104103298, 0.53052538117526306189926779702, 0.67976050878206954495565325186, 1.45960223867565822030913430064, 1.50754710852943044894232065404, 2.23945790092984610411674453970, 2.27124050842289513103068895603, 2.28976696499242668668994722569, 2.65943536581432967175814939287, 2.76833139466050482736431549082, 3.21597190498816595202878816593, 3.32203759262580233174092837570, 3.81876954770496368967671421011, 3.86699181156119072108522408239, 3.91649540821464401928233952970, 4.25637865156061974468331781665, 4.62736487492871232532312959884, 4.83826683885073884470707487387, 5.09977543233331387846820203424, 5.18182255883410808915741099613, 5.20717155187260899914241134519, 5.28093831186049281908105943196, 5.57895694820641188693109378265, 5.62604856124310365688036902192, 6.19445986200286747151343824184

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