Properties

Label 8-1620e4-1.1-c0e4-0-4
Degree $8$
Conductor $6.887\times 10^{12}$
Sign $1$
Analytic cond. $0.427256$
Root an. cond. $0.899158$
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  + 2·7-s − 2·11-s + 4·17-s − 2·23-s + 25-s − 2·31-s − 2·41-s + 2·49-s − 4·53-s + 2·67-s − 4·71-s + 4·73-s − 4·77-s + 2·101-s − 2·103-s + 4·107-s + 2·113-s + 8·119-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + ⋯
L(s)  = 1  + 2·7-s − 2·11-s + 4·17-s − 2·23-s + 25-s − 2·31-s − 2·41-s + 2·49-s − 4·53-s + 2·67-s − 4·71-s + 4·73-s − 4·77-s + 2·101-s − 2·103-s + 4·107-s + 2·113-s + 8·119-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.427256\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.305430952\)
\(L(\frac12)\) \(\approx\) \(1.305430952\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
5$C_2^2$ \( 1 - T^{2} + T^{4} \)
good7$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
13$C_2^3$ \( 1 - T^{4} + T^{8} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
43$C_2^3$ \( 1 - T^{4} + T^{8} \)
47$C_2^3$ \( 1 - T^{4} + T^{8} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
59$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$ \( ( 1 + T + T^{2} )^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2^3$ \( 1 - T^{4} + T^{8} \)
89$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
97$C_2^3$ \( 1 - T^{4} + T^{8} \)
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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

−7.13790900736843269939375503332, −6.56651311369008780948819692533, −6.30795519151329098363724291381, −6.27849522966770459385418135438, −6.05478299350966692998965031741, −5.66744758263255317793636553171, −5.54546492928011380165718243405, −5.33402855952051040018954262106, −5.05587261847517322266761732452, −5.03523700319793670395609716822, −4.94785486995268190644900432953, −4.61489944507902507613442199574, −4.40052075700658960799858091049, −3.85248486766315455387593720460, −3.60193394488654963506184967820, −3.48395694502500866598262326846, −3.45995092856768821334485277478, −3.06355604173032085187074990485, −2.57351230167932303089028828760, −2.53009052315635872044885688369, −2.11681672933630524361109102582, −1.57088094904265804586296582513, −1.50796677308599097424015984281, −1.50602115763110178260499011098, −0.65225055960135622747187196493, 0.65225055960135622747187196493, 1.50602115763110178260499011098, 1.50796677308599097424015984281, 1.57088094904265804586296582513, 2.11681672933630524361109102582, 2.53009052315635872044885688369, 2.57351230167932303089028828760, 3.06355604173032085187074990485, 3.45995092856768821334485277478, 3.48395694502500866598262326846, 3.60193394488654963506184967820, 3.85248486766315455387593720460, 4.40052075700658960799858091049, 4.61489944507902507613442199574, 4.94785486995268190644900432953, 5.03523700319793670395609716822, 5.05587261847517322266761732452, 5.33402855952051040018954262106, 5.54546492928011380165718243405, 5.66744758263255317793636553171, 6.05478299350966692998965031741, 6.27849522966770459385418135438, 6.30795519151329098363724291381, 6.56651311369008780948819692533, 7.13790900736843269939375503332

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