Properties

Label 8-1200e4-1.1-c1e4-0-22
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 12·7-s + 8·9-s − 48·21-s − 12·27-s + 8·31-s + 8·37-s − 8·43-s + 72·49-s − 24·61-s + 96·63-s + 16·67-s − 12·73-s + 23·81-s − 32·93-s + 52·97-s − 44·103-s − 32·111-s + 40·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s − 288·147-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2.30·3-s + 4.53·7-s + 8/3·9-s − 10.4·21-s − 2.30·27-s + 1.43·31-s + 1.31·37-s − 1.21·43-s + 72/7·49-s − 3.07·61-s + 12.0·63-s + 1.95·67-s − 1.40·73-s + 23/9·81-s − 3.31·93-s + 5.27·97-s − 4.33·103-s − 3.03·111-s + 3.63·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 23.7·147-s + 0.0819·149-s + 0.0813·151-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

\(L(1)\) \(\approx\) \(3.400046477\)
\(L(\frac12)\) \(\approx\) \(3.400046477\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 158 T^{4} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 3518 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 4174 T^{4} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 8722 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 - 8 T + p T^{2} )^{2} \)
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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

−7.06829887624432967768639841045, −6.45808988721811816930781415782, −6.37803349088814428252209582392, −6.36195223009988681703948942208, −6.15222279721812443143147545511, −5.69671394688587711301125808722, −5.52913630451042555307627608728, −5.37471974882660197280746497393, −5.01433375717286440336004035256, −4.97159001585485841619855820413, −4.92568993109496884404957976731, −4.55563619792721296359389216392, −4.48593962508274016157016990480, −4.11467524151267280266728841009, −4.04325814336621241718168657066, −3.67316399496878847773510265373, −2.98479601770405182424978645371, −2.91608857919995405952627524945, −2.50262860453763827825448674382, −1.81345319030841354769446526460, −1.79541117466357769325734428489, −1.75324699816212045526266982568, −1.27530577881669298596963275819, −0.75986099135028635278330319516, −0.61355404920622851044851102891, 0.61355404920622851044851102891, 0.75986099135028635278330319516, 1.27530577881669298596963275819, 1.75324699816212045526266982568, 1.79541117466357769325734428489, 1.81345319030841354769446526460, 2.50262860453763827825448674382, 2.91608857919995405952627524945, 2.98479601770405182424978645371, 3.67316399496878847773510265373, 4.04325814336621241718168657066, 4.11467524151267280266728841009, 4.48593962508274016157016990480, 4.55563619792721296359389216392, 4.92568993109496884404957976731, 4.97159001585485841619855820413, 5.01433375717286440336004035256, 5.37471974882660197280746497393, 5.52913630451042555307627608728, 5.69671394688587711301125808722, 6.15222279721812443143147545511, 6.36195223009988681703948942208, 6.37803349088814428252209582392, 6.45808988721811816930781415782, 7.06829887624432967768639841045

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