Properties

Label 8-3584e4-1.1-c0e4-0-4
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

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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

\(L(\frac{1}{2})\) \(\approx\) \(2.334655283\)
\(L(\frac12)\) \(\approx\) \(2.334655283\)
\(L(1)\) 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

−6.41308578932996328607184793168, −5.89318797626734875488982857651, −5.82334399049606068110052591746, −5.66895450037369079482594448878, −5.54905046090371344498692225880, −5.26601461054313002469683555138, −4.87983044166724496821516152477, −4.85305527769848657169341325760, −4.84090532637585864881173315109, −4.26690851059388906974143169980, −4.19410894358792032749728721280, −3.95926916136639188772365098370, −3.74922575953610651330772104277, −3.52202474950726525188352021695, −3.38597114131175002917153142418, −3.23205804989832047987685988740, −3.14739466895736509218870991284, −2.70305810637319658797469385740, −2.09369277184077846606514369400, −2.05716030755713228218105476889, −1.90039386967745192039866151846, −1.33818556008429459999772826554, −1.31980013440966420702442776858, −1.24491026419306281339092878023, −0.57184627605194205377383849911, 0.57184627605194205377383849911, 1.24491026419306281339092878023, 1.31980013440966420702442776858, 1.33818556008429459999772826554, 1.90039386967745192039866151846, 2.05716030755713228218105476889, 2.09369277184077846606514369400, 2.70305810637319658797469385740, 3.14739466895736509218870991284, 3.23205804989832047987685988740, 3.38597114131175002917153142418, 3.52202474950726525188352021695, 3.74922575953610651330772104277, 3.95926916136639188772365098370, 4.19410894358792032749728721280, 4.26690851059388906974143169980, 4.84090532637585864881173315109, 4.85305527769848657169341325760, 4.87983044166724496821516152477, 5.26601461054313002469683555138, 5.54905046090371344498692225880, 5.66895450037369079482594448878, 5.82334399049606068110052591746, 5.89318797626734875488982857651, 6.41308578932996328607184793168

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