Properties

Label 8-1200e4-1.1-c1e4-0-2
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $8430.11$
Root an. cond. $3.09548$
Motivic weight $1$
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  − 24·17-s − 16·49-s + 24·53-s + 32·61-s − 9·81-s − 16·109-s − 24·113-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 5.82·17-s − 2.28·49-s + 3.29·53-s + 4.09·61-s − 81-s − 1.53·109-s − 2.25·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(8430.11\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1942549151\)
\(L(\frac12)\) \(\approx\) \(0.1942549151\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.96473337604456935519266085216, −6.93372253683325611998882820066, −6.69401745349335308892127954684, −6.35092770901810737144985724633, −6.08046694615787282544863136460, −6.01938619768155052776068019111, −5.62172684328578717489952923136, −5.37664337051034007770040203822, −5.12365678922605946968864091330, −4.83045740513117562751330629793, −4.71745410487922287316781359951, −4.45400378971733279803856660160, −4.21852707496562364812517222161, −3.93198195935572767804944355242, −3.83138794626566215938153338500, −3.69442272685433421924350737668, −2.92880070839650173753349054373, −2.90428125376114398812313704798, −2.37794954841143723073264539793, −2.34337653681297377569683665906, −2.05388921748437615148930135847, −1.92327126218442408510220523927, −1.28989475626296779572636178527, −0.76161168520116107661676695950, −0.10926304072506061458286929194, 0.10926304072506061458286929194, 0.76161168520116107661676695950, 1.28989475626296779572636178527, 1.92327126218442408510220523927, 2.05388921748437615148930135847, 2.34337653681297377569683665906, 2.37794954841143723073264539793, 2.90428125376114398812313704798, 2.92880070839650173753349054373, 3.69442272685433421924350737668, 3.83138794626566215938153338500, 3.93198195935572767804944355242, 4.21852707496562364812517222161, 4.45400378971733279803856660160, 4.71745410487922287316781359951, 4.83045740513117562751330629793, 5.12365678922605946968864091330, 5.37664337051034007770040203822, 5.62172684328578717489952923136, 6.01938619768155052776068019111, 6.08046694615787282544863136460, 6.35092770901810737144985724633, 6.69401745349335308892127954684, 6.93372253683325611998882820066, 6.96473337604456935519266085216

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