Properties

Label 8-600e4-1.1-c1e4-0-16
Degree $8$
Conductor $129600000000$
Sign $1$
Analytic cond. $526.882$
Root an. cond. $2.18884$
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 + 8·7-s + 8·9-s + 32·21-s + 12·27-s + 32·31-s − 32·37-s + 8·43-s + 32·49-s − 24·61-s + 64·63-s + 24·67-s − 32·73-s + 23·81-s + 128·93-s + 32·97-s + 24·103-s − 128·111-s − 20·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s + 128·147-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2.30·3-s + 3.02·7-s + 8/3·9-s + 6.98·21-s + 2.30·27-s + 5.74·31-s − 5.26·37-s + 1.21·43-s + 32/7·49-s − 3.07·61-s + 8.06·63-s + 2.93·67-s − 3.74·73-s + 23/9·81-s + 13.2·93-s + 3.24·97-s + 2.36·103-s − 12.1·111-s − 1.81·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.5·147-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(526.882\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.01171362\)
\(L(\frac12)\) \(\approx\) \(12.01171362\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 254 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 958 T^{4} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
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$C_2^3$ \( 1 + 3682 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 3854 T^{4} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2^3$ \( 1 - 12878 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.81375719714500017868336676411, −7.54434022681102129259711095366, −7.27331507502432617979734143967, −7.25048871863829031643520472450, −6.99183340137787615935079542034, −6.43561408897137392191719354693, −6.32615977138206328877332237972, −6.12257249549774326151727324542, −5.77765085315488845289309526320, −5.27487929781584120190708097856, −4.93579949982330244563366460651, −4.87254526941122579339200741349, −4.81949693080983703297922330155, −4.47885609832175514594065076086, −4.22177179584801935656379819514, −3.85038367217096879344964794393, −3.42898663251707303857578393591, −3.34470924901840740743919297413, −2.86690011280739985368394662212, −2.57669095875863565541452278991, −2.43534831154197271873436270643, −1.85214976429759715755345088730, −1.75838974038615239535563466949, −1.31086555829133166954700244857, −0.936459502804013404569416563802, 0.936459502804013404569416563802, 1.31086555829133166954700244857, 1.75838974038615239535563466949, 1.85214976429759715755345088730, 2.43534831154197271873436270643, 2.57669095875863565541452278991, 2.86690011280739985368394662212, 3.34470924901840740743919297413, 3.42898663251707303857578393591, 3.85038367217096879344964794393, 4.22177179584801935656379819514, 4.47885609832175514594065076086, 4.81949693080983703297922330155, 4.87254526941122579339200741349, 4.93579949982330244563366460651, 5.27487929781584120190708097856, 5.77765085315488845289309526320, 6.12257249549774326151727324542, 6.32615977138206328877332237972, 6.43561408897137392191719354693, 6.99183340137787615935079542034, 7.25048871863829031643520472450, 7.27331507502432617979734143967, 7.54434022681102129259711095366, 7.81375719714500017868336676411

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