Properties

Label 8-280e4-1.1-c1e4-0-3
Degree $8$
Conductor $6146560000$
Sign $1$
Analytic cond. $24.9885$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 6·3-s + 2·4-s + 2·5-s − 12·6-s + 4·8-s + 15·9-s + 4·10-s − 4·11-s − 12·12-s + 12·13-s − 12·15-s + 8·16-s + 6·17-s + 30·18-s − 18·19-s + 4·20-s − 8·22-s + 24·23-s − 24·24-s + 25-s + 24·26-s − 18·27-s − 24·30-s − 6·31-s + 8·32-s + 24·33-s + ⋯
L(s)  = 1  + 1.41·2-s − 3.46·3-s + 4-s + 0.894·5-s − 4.89·6-s + 1.41·8-s + 5·9-s + 1.26·10-s − 1.20·11-s − 3.46·12-s + 3.32·13-s − 3.09·15-s + 2·16-s + 1.45·17-s + 7.07·18-s − 4.12·19-s + 0.894·20-s − 1.70·22-s + 5.00·23-s − 4.89·24-s + 1/5·25-s + 4.70·26-s − 3.46·27-s − 4.38·30-s − 1.07·31-s + 1.41·32-s + 4.17·33-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(24.9885\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.471181668\)
\(L(\frac12)\) \(\approx\) \(1.471181668\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
11$D_4\times C_2$ \( 1 + 4 T + 2 T^{2} - 32 T^{3} - 101 T^{4} - 32 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 6 T + 40 T^{2} - 168 T^{3} + 699 T^{4} - 168 p T^{5} + 40 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 18 T + 164 T^{2} + 1008 T^{3} + 4827 T^{4} + 1008 p T^{5} + 164 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 24 T + 285 T^{2} - 2232 T^{3} + 12536 T^{4} - 2232 p T^{5} + 285 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 6 T^{2} - 661 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 6 T - 8 T^{2} - 108 T^{3} - 141 T^{4} - 108 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
41$C_2^2$ \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 12 T + 26 T^{2} - 288 T^{3} + 4947 T^{4} - 288 p T^{5} + 26 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 18 T + 216 T^{2} - 1944 T^{3} + 14579 T^{4} - 1944 p T^{5} + 216 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 6 T + 124 T^{2} - 672 T^{3} + 9771 T^{4} - 672 p T^{5} + 124 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 12 T - 11 T^{2} - 396 T^{3} + 11520 T^{4} - 396 p T^{5} - 11 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 2 T - 23 T^{2} - 214 T^{3} - 4028 T^{4} - 214 p T^{5} - 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 180 T^{2} + 17594 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 30 T + 512 T^{2} + 6360 T^{3} + 61515 T^{4} + 6360 p T^{5} + 512 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 6 T + 164 T^{2} + 912 T^{3} + 17811 T^{4} + 912 p T^{5} + 164 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 110 T^{2} + 6003 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 30 T + 517 T^{2} - 6510 T^{3} + 65868 T^{4} - 6510 p T^{5} + 517 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \)
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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

−8.791728144548701526497460815535, −8.224070132457629828057656207654, −8.167508113869454100921636800297, −7.936188081983036929847304088329, −7.16189337562416938943674059295, −7.00230796136621082639941624582, −6.88099272370861388992834242266, −6.56792973664700916856743705831, −6.43450984086770305383163460526, −5.95454125217632538908522248559, −5.84353752345970497611552837095, −5.70470497488294204251490020701, −5.40807451643272543316684187045, −5.31148019118832097518367545871, −5.02006561977293395249861964572, −4.78262969922921196161621880485, −4.33069378762928077674375580416, −4.04882028821424225776011425544, −3.77049524703701860951555729538, −3.22754178116673726826814580947, −2.98656674267563331142827422427, −2.35168056102284378835320241292, −1.65275744805332902980747983456, −1.25241552587241277003225358824, −0.67892632169772755405947824905, 0.67892632169772755405947824905, 1.25241552587241277003225358824, 1.65275744805332902980747983456, 2.35168056102284378835320241292, 2.98656674267563331142827422427, 3.22754178116673726826814580947, 3.77049524703701860951555729538, 4.04882028821424225776011425544, 4.33069378762928077674375580416, 4.78262969922921196161621880485, 5.02006561977293395249861964572, 5.31148019118832097518367545871, 5.40807451643272543316684187045, 5.70470497488294204251490020701, 5.84353752345970497611552837095, 5.95454125217632538908522248559, 6.43450984086770305383163460526, 6.56792973664700916856743705831, 6.88099272370861388992834242266, 7.00230796136621082639941624582, 7.16189337562416938943674059295, 7.936188081983036929847304088329, 8.167508113869454100921636800297, 8.224070132457629828057656207654, 8.791728144548701526497460815535

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