Properties

Label 8-4608e4-1.1-c1e4-0-23
Degree $8$
Conductor $4.509\times 10^{14}$
Sign $1$
Analytic cond. $1.83298\times 10^{6}$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 16·17-s − 8·25-s + 16·41-s − 24·49-s − 16·73-s + 64·89-s + 24·97-s + 32·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.88·17-s − 8/5·25-s + 2.49·41-s − 3.42·49-s − 1.87·73-s + 6.78·89-s + 2.43·97-s + 3.01·113-s − 1.81·121-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 − 0.307·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.83298\times 10^{6}\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.141952539\)
\(L(\frac12)\) \(\approx\) \(6.141952539\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 108 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
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

−5.99099403495706668955644562406, −5.75847813225027814418201801914, −5.44897765612470604599926325530, −5.39964650025734038334357711465, −5.05255637025177608125188705439, −4.79909383224258825004834430031, −4.75628857675733656447259505000, −4.59351529206510630464184421610, −4.43220427854209321098822670179, −3.88075901163517833051891350768, −3.75650517247086482236838000980, −3.60831019286873278136412371419, −3.59185758374375663150232920207, −3.20853513645789752437782973557, −3.03213850573840251631017500889, −2.88199546820798744855326852443, −2.73750185431110382328593866208, −2.13011582929890769845634910224, −1.94613753697404705061423507002, −1.80922908077163399303214316143, −1.73437906404678433804948375584, −1.04727951821994433156680124642, −0.961092984726175068772015173827, −0.73970149342367046213149823429, −0.35521274087390964463913027938, 0.35521274087390964463913027938, 0.73970149342367046213149823429, 0.961092984726175068772015173827, 1.04727951821994433156680124642, 1.73437906404678433804948375584, 1.80922908077163399303214316143, 1.94613753697404705061423507002, 2.13011582929890769845634910224, 2.73750185431110382328593866208, 2.88199546820798744855326852443, 3.03213850573840251631017500889, 3.20853513645789752437782973557, 3.59185758374375663150232920207, 3.60831019286873278136412371419, 3.75650517247086482236838000980, 3.88075901163517833051891350768, 4.43220427854209321098822670179, 4.59351529206510630464184421610, 4.75628857675733656447259505000, 4.79909383224258825004834430031, 5.05255637025177608125188705439, 5.39964650025734038334357711465, 5.44897765612470604599926325530, 5.75847813225027814418201801914, 5.99099403495706668955644562406

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