Properties

Label 8-1200e4-1.1-c1e4-0-25
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 $4$

Origins

Origins of factors

Downloads

Learn more

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

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

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: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.33141066806559101920320673994, −7.08865658340810206136314088578, −6.90312460301925017665630973448, −6.64770209205872827267057240311, −6.59456570505574244109479947746, −6.14338251429471009493355995330, −5.98950487083743241321939682913, −5.95040283240010447057077064634, −5.79472229497362126952728517986, −5.47889022127029328100287014697, −5.16720725357775466167810705883, −5.10517222830260620554518905651, −4.95349925205421325153156435037, −4.60051738945155253587241994299, −4.08462660640692026455977735693, −3.91058196048793774274400447395, −3.59892778277536946967825673883, −3.54269177073260169902359654027, −3.32330537809552337108992570988, −2.95623639703052422194080173491, −2.88691164004279061749837348432, −2.08122662945271656368633171479, −1.83313232375084836687615780898, −1.55996913568475495543508443069, −1.33915074520200397122627638469, 0, 0, 0, 0, 1.33915074520200397122627638469, 1.55996913568475495543508443069, 1.83313232375084836687615780898, 2.08122662945271656368633171479, 2.88691164004279061749837348432, 2.95623639703052422194080173491, 3.32330537809552337108992570988, 3.54269177073260169902359654027, 3.59892778277536946967825673883, 3.91058196048793774274400447395, 4.08462660640692026455977735693, 4.60051738945155253587241994299, 4.95349925205421325153156435037, 5.10517222830260620554518905651, 5.16720725357775466167810705883, 5.47889022127029328100287014697, 5.79472229497362126952728517986, 5.95040283240010447057077064634, 5.98950487083743241321939682913, 6.14338251429471009493355995330, 6.59456570505574244109479947746, 6.64770209205872827267057240311, 6.90312460301925017665630973448, 7.08865658340810206136314088578, 7.33141066806559101920320673994

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