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Properties

Label 8-2016e4-1.1-c0e4-0-4
Degree $8$
Conductor $1.652\times 10^{13}$
Sign $1$
Analytic cond. $1.02468$
Root an. cond. $1.00305$
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 − 16-s − 4·23-s + 4·29-s + 4·43-s + 4·67-s − 4·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4·11-s − 16-s − 4·23-s + 4·29-s + 4·43-s + 4·67-s − 4·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.02468\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7148319394\)
\(L(\frac12)\) \(\approx\) \(0.7148319394\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good5$C_4\times C_2$ \( 1 + T^{8} \)
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_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
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_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
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

−6.61777728932487910636892613954, −6.46771690879275953198131462458, −6.19545467863175574082280317763, −6.09597211860803420946915579794, −5.85251563289449660817826043711, −5.70885476744497502414113534925, −5.34502549523002123212430684397, −5.27605706992106107318859452454, −5.01111043640053915776577029886, −4.77317248135290003602464110665, −4.76589922692891920454601728914, −4.17403593939588406101064477568, −4.11347635101029494158171089117, −3.96738977559825674273755082752, −3.94798783568929379133448673431, −3.06199598888594220859908858667, −3.05346619041359283389341239662, −2.92296525084686671431757414228, −2.41404507905652270447695420645, −2.41015333456661254581762328876, −2.25615976073609548856306388821, −2.05677794648914722140594652200, −1.50608241021487154035623972960, −0.77558862603944845971577500018, −0.55347212610948325143440447875, 0.55347212610948325143440447875, 0.77558862603944845971577500018, 1.50608241021487154035623972960, 2.05677794648914722140594652200, 2.25615976073609548856306388821, 2.41015333456661254581762328876, 2.41404507905652270447695420645, 2.92296525084686671431757414228, 3.05346619041359283389341239662, 3.06199598888594220859908858667, 3.94798783568929379133448673431, 3.96738977559825674273755082752, 4.11347635101029494158171089117, 4.17403593939588406101064477568, 4.76589922692891920454601728914, 4.77317248135290003602464110665, 5.01111043640053915776577029886, 5.27605706992106107318859452454, 5.34502549523002123212430684397, 5.70885476744497502414113534925, 5.85251563289449660817826043711, 6.09597211860803420946915579794, 6.19545467863175574082280317763, 6.46771690879275953198131462458, 6.61777728932487910636892613954

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