↑

Properties

Label 8-300e4-1.1-c0e4-0-0
Degree $8$
Conductor $8100000000$
Sign $1$
Analytic cond. $0.000502473$
Root an. cond. $0.386936$
Motivic weight $0$
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-s − 8·61-s − 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  − 16-s − 8·61-s − 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.000502473\)
Root analytic conductor: \(0.386936\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3512582598\)
\(L(\frac12)\) \(\approx\) \(0.3512582598\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_1$ \( ( 1 + T )^{8} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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

−9.043328147972449035216947636172, −8.432854045302240909782223598137, −8.411918217483788657385467673099, −7.959230405478041662023382150220, −7.63682054159941470575817119598, −7.59317125952758812395903266448, −7.39702987660357997151345068660, −7.05624517925892448833598394965, −6.57414297771465698683497511585, −6.41489599425655357514547952408, −6.24012661431642841373249473491, −6.08961050487577342783472551019, −5.58502092796892536183335039452, −5.20514727985965618683528124107, −5.16295324392928414068576154021, −4.67850518427808979324199323128, −4.36061144814064941935865519036, −4.11189807717108522972642807070, −4.05799487665792761554002855629, −3.10217914592319341287474613947, −3.08218222097724472371337771954, −2.95202118631667417301243876022, −2.27254191941686987914538653412, −1.64326021482888890759658743212, −1.53297601872577559851715814642, 1.53297601872577559851715814642, 1.64326021482888890759658743212, 2.27254191941686987914538653412, 2.95202118631667417301243876022, 3.08218222097724472371337771954, 3.10217914592319341287474613947, 4.05799487665792761554002855629, 4.11189807717108522972642807070, 4.36061144814064941935865519036, 4.67850518427808979324199323128, 5.16295324392928414068576154021, 5.20514727985965618683528124107, 5.58502092796892536183335039452, 6.08961050487577342783472551019, 6.24012661431642841373249473491, 6.41489599425655357514547952408, 6.57414297771465698683497511585, 7.05624517925892448833598394965, 7.39702987660357997151345068660, 7.59317125952758812395903266448, 7.63682054159941470575817119598, 7.959230405478041662023382150220, 8.411918217483788657385467673099, 8.432854045302240909782223598137, 9.043328147972449035216947636172

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