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Properties

Label 8-1400e4-1.1-c0e4-0-4
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $0.238309$
Root an. cond. $0.835877$
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-s + 3·3-s − 5-s − 3·6-s + 4·7-s + 6·9-s + 10-s − 2·13-s − 4·14-s − 3·15-s − 6·18-s − 2·19-s + 12·21-s − 2·23-s + 2·26-s + 10·27-s + 3·30-s + 32-s − 4·35-s + 2·38-s − 6·39-s − 12·42-s − 6·45-s + 2·46-s + 10·49-s − 10·54-s − 6·57-s + ⋯
L(s)  = 1  − 2-s + 3·3-s − 5-s − 3·6-s + 4·7-s + 6·9-s + 10-s − 2·13-s − 4·14-s − 3·15-s − 6·18-s − 2·19-s + 12·21-s − 2·23-s + 2·26-s + 10·27-s + 3·30-s + 32-s − 4·35-s + 2·38-s − 6·39-s − 12·42-s − 6·45-s + 2·46-s + 10·49-s − 10·54-s − 6·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.238309\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.246004695\)
\(L(\frac12)\) \(\approx\) \(2.246004695\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
good3$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
71$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−7.31323044258625766981827533383, −7.00123232752279273750761110073, −6.85024891445701741431447899067, −6.81778327730881742997899223889, −6.23826692587339069351911449873, −5.92823639396485152926069495388, −5.76596059249084394638027910492, −5.13968165364519432561176670455, −5.04277213925155067510430901347, −5.01154292528578222179657269558, −4.72790293626288624988884649695, −4.44251355265355513159105048187, −4.28570192734059590896901041333, −3.99961016097886550183992273094, −3.90464431133248096813380996188, −3.65088752799370993645319542647, −3.63511377884825734187238321510, −2.63002491771269250211801908806, −2.48721480605046224601032165137, −2.40163299793942516562535716191, −2.38912255788269706732813757278, −2.03560751753510727290825106473, −1.61389730590388008597355224516, −1.36242121848083035217948115954, −0.999118085833443317557835040990, 0.999118085833443317557835040990, 1.36242121848083035217948115954, 1.61389730590388008597355224516, 2.03560751753510727290825106473, 2.38912255788269706732813757278, 2.40163299793942516562535716191, 2.48721480605046224601032165137, 2.63002491771269250211801908806, 3.63511377884825734187238321510, 3.65088752799370993645319542647, 3.90464431133248096813380996188, 3.99961016097886550183992273094, 4.28570192734059590896901041333, 4.44251355265355513159105048187, 4.72790293626288624988884649695, 5.01154292528578222179657269558, 5.04277213925155067510430901347, 5.13968165364519432561176670455, 5.76596059249084394638027910492, 5.92823639396485152926069495388, 6.23826692587339069351911449873, 6.81778327730881742997899223889, 6.85024891445701741431447899067, 7.00123232752279273750761110073, 7.31323044258625766981827533383

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