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Properties

Label 8-3584e4-1.1-c0e4-0-5
Degree $8$
Conductor $1.650\times 10^{14}$
Sign $1$
Analytic cond. $10.2352$
Root an. cond. $1.33740$
Motivic weight $0$
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  − 4·11-s + 4·23-s + 4·29-s + 4·37-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 4·11-s + 4·23-s + 4·29-s + 4·37-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(10.2352\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.662507534\)
\(L(\frac12)\) \(\approx\) \(1.662507534\)
\(L(1)\) 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 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.32365705017853690458337977366, −5.94241032697833497107627595968, −5.70175724244050436504559984286, −5.68044293368792910208219139194, −5.56035183062861750494834733957, −5.00603185741370927174777646266, −4.91822391439632280976716711426, −4.88521595745778831048057574183, −4.85959249557482080170699193665, −4.46299622357295447609982929766, −4.45390107142037661639832817145, −4.18806346514721643822110338182, −3.72685142504968417407521490912, −3.26961865642575345354757431426, −3.14023172761004517891725750238, −3.10650199473350869445964997173, −2.89618446345695699561317847539, −2.65257611088873913911994785542, −2.35718319029670465618172548980, −2.30252273172149934569813484626, −2.26340094068151281424144182679, −1.30207986445833854336998356243, −1.20205910977634357760390577406, −0.867640316064430770546928549890, −0.62417873863026576592324344489, 0.62417873863026576592324344489, 0.867640316064430770546928549890, 1.20205910977634357760390577406, 1.30207986445833854336998356243, 2.26340094068151281424144182679, 2.30252273172149934569813484626, 2.35718319029670465618172548980, 2.65257611088873913911994785542, 2.89618446345695699561317847539, 3.10650199473350869445964997173, 3.14023172761004517891725750238, 3.26961865642575345354757431426, 3.72685142504968417407521490912, 4.18806346514721643822110338182, 4.45390107142037661639832817145, 4.46299622357295447609982929766, 4.85959249557482080170699193665, 4.88521595745778831048057574183, 4.91822391439632280976716711426, 5.00603185741370927174777646266, 5.56035183062861750494834733957, 5.68044293368792910208219139194, 5.70175724244050436504559984286, 5.94241032697833497107627595968, 6.32365705017853690458337977366

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