Properties

Label 8-476e4-1.1-c0e4-0-0
Degree $8$
Conductor $51336683776$
Sign $1$
Analytic cond. $0.00318461$
Root an. cond. $0.487396$
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  + 2·2-s + 4-s − 2·8-s − 9-s + 4·13-s − 4·16-s − 2·17-s − 2·18-s − 2·25-s + 8·26-s − 2·32-s − 4·34-s − 36-s − 2·49-s − 4·50-s + 4·52-s + 2·53-s + 3·64-s − 2·68-s + 2·72-s + 81-s − 2·89-s − 4·98-s − 2·100-s + 4·101-s − 8·104-s + 4·106-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 2·8-s − 9-s + 4·13-s − 4·16-s − 2·17-s − 2·18-s − 2·25-s + 8·26-s − 2·32-s − 4·34-s − 36-s − 2·49-s − 4·50-s + 4·52-s + 2·53-s + 3·64-s − 2·68-s + 2·72-s + 81-s − 2·89-s − 4·98-s − 2·100-s + 4·101-s − 8·104-s + 4·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 7^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(0.00318461\)
Root analytic conductor: \(0.487396\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 7^{4} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024725000\)
\(L(\frac12)\) \(\approx\) \(1.024725000\)
\(L(1)\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T + T^{2} )^{2} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
13$C_2$ \( ( 1 - T + T^{2} )^{4} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
71$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
79$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
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

−8.295134243651476894657254720512, −8.111145523124783815237780202687, −7.79619662493473746985943868253, −7.43124846454260907082139745504, −6.91102574826430035000289349049, −6.81473999656473342397938340318, −6.47252090941402724136107794406, −6.28913316104520850601139058727, −6.00991797545547029172835080837, −5.92906387769723280787642798314, −5.91439906930685071175477783227, −5.29448475235018583592176983778, −5.27576068283942987298436104519, −4.97159460935988884613838754383, −4.43083053375474411949538090420, −4.24242304153068494756273286680, −4.01465038350638189169196351049, −3.97962228810159768289381329074, −3.39453644134763664225732626843, −3.32401147816787240906483124784, −3.19847471020486908205697602242, −2.54572849399032149803958116538, −2.17144802826572071518473476174, −1.85648996883229754103561318628, −1.08987278157440062378599310698, 1.08987278157440062378599310698, 1.85648996883229754103561318628, 2.17144802826572071518473476174, 2.54572849399032149803958116538, 3.19847471020486908205697602242, 3.32401147816787240906483124784, 3.39453644134763664225732626843, 3.97962228810159768289381329074, 4.01465038350638189169196351049, 4.24242304153068494756273286680, 4.43083053375474411949538090420, 4.97159460935988884613838754383, 5.27576068283942987298436104519, 5.29448475235018583592176983778, 5.91439906930685071175477783227, 5.92906387769723280787642798314, 6.00991797545547029172835080837, 6.28913316104520850601139058727, 6.47252090941402724136107794406, 6.81473999656473342397938340318, 6.91102574826430035000289349049, 7.43124846454260907082139745504, 7.79619662493473746985943868253, 8.111145523124783815237780202687, 8.295134243651476894657254720512

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