Properties

Label 8-1260e4-1.1-c1e4-0-0
Degree $8$
Conductor $2.520\times 10^{12}$
Sign $1$
Analytic cond. $10246.8$
Root an. cond. $3.17193$
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  + 2·5-s + 2·11-s + 2·17-s − 6·19-s − 2·23-s + 25-s − 20·29-s − 2·31-s + 4·37-s + 12·41-s + 8·43-s + 12·47-s − 7·49-s − 4·53-s + 4·55-s + 6·59-s − 16·61-s + 4·67-s − 44·71-s + 6·79-s + 12·83-s + 4·85-s − 2·89-s − 12·95-s + 32·97-s + 6·101-s − 24·103-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s + 0.485·17-s − 1.37·19-s − 0.417·23-s + 1/5·25-s − 3.71·29-s − 0.359·31-s + 0.657·37-s + 1.87·41-s + 1.21·43-s + 1.75·47-s − 49-s − 0.549·53-s + 0.539·55-s + 0.781·59-s − 2.04·61-s + 0.488·67-s − 5.22·71-s + 0.675·79-s + 1.31·83-s + 0.433·85-s − 0.211·89-s − 1.23·95-s + 3.24·97-s + 0.597·101-s − 2.36·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(10246.8\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1260} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5656197542\)
\(L(\frac12)\) \(\approx\) \(0.5656197542\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
good11$D_4\times C_2$ \( 1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 12 p T^{5} - 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 12 p T^{5} - 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 6 T + 17 T^{2} - 6 p T^{3} - 36 p T^{4} - 6 p^{2} T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 12 p T^{5} - 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 10 T + 76 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 2 T - p T^{2} - 54 T^{3} + 140 T^{4} - 54 p T^{5} - p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 4 T - 55 T^{2} + 12 T^{3} + 3080 T^{4} + 12 p T^{5} - 55 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 4 T - 66 T^{2} - 96 T^{3} + 3067 T^{4} - 96 p T^{5} - 66 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 6 T - 28 T^{2} + 324 T^{3} - 1509 T^{4} + 324 p T^{5} - 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 4 T - 115 T^{2} + 12 T^{3} + 11600 T^{4} + 12 p T^{5} - 115 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 22 T + 256 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 + 29 T^{2} - 4488 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 6 T - 103 T^{2} + 114 T^{3} + 10236 T^{4} + 114 p T^{5} - 103 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 2 T - 348 T^{3} - 8261 T^{4} - 348 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
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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

−6.99055892359695165805950821657, −6.61889529485795986571443313741, −6.25347578913483579641513818144, −6.15674738943037634945185639912, −6.10183484186941932344513029080, −5.85417019853883451977584433147, −5.79417144096410689711830311289, −5.44019881541301818436715284472, −5.13700009680785864729033428788, −4.98300139744358199608984380729, −4.65858099046749551853078638757, −4.28497816259393512866648339923, −4.16038531675518489587041313879, −3.95871108753831080156254824841, −3.79662224238735448461526805623, −3.48873397837469220704955005331, −3.04959555633982764802269314723, −2.84095122546039708125001086604, −2.57417255265591881432418633395, −2.09667361890457538878958205230, −1.99080297305571502335173203087, −1.79725246722893934108647973852, −1.22800476100533845974334487086, −1.05728629360491220302824439036, −0.14719914391265590872760251820, 0.14719914391265590872760251820, 1.05728629360491220302824439036, 1.22800476100533845974334487086, 1.79725246722893934108647973852, 1.99080297305571502335173203087, 2.09667361890457538878958205230, 2.57417255265591881432418633395, 2.84095122546039708125001086604, 3.04959555633982764802269314723, 3.48873397837469220704955005331, 3.79662224238735448461526805623, 3.95871108753831080156254824841, 4.16038531675518489587041313879, 4.28497816259393512866648339923, 4.65858099046749551853078638757, 4.98300139744358199608984380729, 5.13700009680785864729033428788, 5.44019881541301818436715284472, 5.79417144096410689711830311289, 5.85417019853883451977584433147, 6.10183484186941932344513029080, 6.15674738943037634945185639912, 6.25347578913483579641513818144, 6.61889529485795986571443313741, 6.99055892359695165805950821657

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