Properties

Label 8-2001e4-1.1-c0e4-0-1
Degree $8$
Conductor $1.603\times 10^{13}$
Sign $1$
Analytic cond. $0.994527$
Root an. cond. $0.999314$
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 + 2·4-s + 9-s − 3·16-s − 2·18-s + 4·25-s + 2·31-s + 4·32-s + 2·36-s + 2·41-s − 2·47-s + 4·49-s − 8·50-s − 4·62-s − 2·64-s + 2·73-s − 4·82-s + 4·94-s − 8·98-s + 8·100-s − 4·101-s + 4·124-s + 127-s − 2·128-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2·2-s + 2·4-s + 9-s − 3·16-s − 2·18-s + 4·25-s + 2·31-s + 4·32-s + 2·36-s + 2·41-s − 2·47-s + 4·49-s − 8·50-s − 4·62-s − 2·64-s + 2·73-s − 4·82-s + 4·94-s − 8·98-s + 8·100-s − 4·101-s + 4·124-s + 127-s − 2·128-s + 131-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 23^{4} \cdot 29^{4}\)
Sign: $1$
Analytic conductor: \(0.994527\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 23^{4} \cdot 29^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7283346700\)
\(L(\frac12)\) \(\approx\) \(0.7283346700\)
\(L(1)\) 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)$
bad3$C_2^2$ \( 1 - T^{2} + T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
good2$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−6.74931738114314809733842765572, −6.70254386874767697018534394739, −6.62678357677016197280100112272, −6.16787685037953061378513476204, −6.02092574292142631439586123884, −5.69867129524085286195481193168, −5.45284178150281076795511102209, −5.22513946429756488004515141268, −4.93061985556174955312500072481, −4.71053927755874912365606573989, −4.63429578908161776856387020092, −4.35922411813746182554232787230, −4.15565750339888851861985690707, −4.08342838137541537382530050618, −3.46677617296152623034344502172, −3.39657362086873662715205391487, −3.06314956301727082962698382275, −2.61322163691978724226645359645, −2.41534826216741107223804090846, −2.38730044178192699650682811334, −2.05922790361499140130650471231, −1.37099445475855887632262427722, −1.29856665633053463670349467816, −0.915726320461567854752598713572, −0.915586976546599552619892365333, 0.915586976546599552619892365333, 0.915726320461567854752598713572, 1.29856665633053463670349467816, 1.37099445475855887632262427722, 2.05922790361499140130650471231, 2.38730044178192699650682811334, 2.41534826216741107223804090846, 2.61322163691978724226645359645, 3.06314956301727082962698382275, 3.39657362086873662715205391487, 3.46677617296152623034344502172, 4.08342838137541537382530050618, 4.15565750339888851861985690707, 4.35922411813746182554232787230, 4.63429578908161776856387020092, 4.71053927755874912365606573989, 4.93061985556174955312500072481, 5.22513946429756488004515141268, 5.45284178150281076795511102209, 5.69867129524085286195481193168, 6.02092574292142631439586123884, 6.16787685037953061378513476204, 6.62678357677016197280100112272, 6.70254386874767697018534394739, 6.74931738114314809733842765572

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