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Properties

Label 8-1260e4-1.1-c0e4-0-6
Degree $8$
Conductor $2.520\times 10^{12}$
Sign $1$
Analytic cond. $0.156354$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 16-s − 4·17-s + 10·25-s + 4·37-s − 4·80-s − 16·85-s + 20·125-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 + 16·185-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4·5-s − 16-s − 4·17-s + 10·25-s + 4·37-s − 4·80-s − 16·85-s + 20·125-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 + 16·185-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.156354\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.195258527\)
\(L(\frac12)\) \(\approx\) \(2.195258527\)
\(L(1)\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
good11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
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$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−6.82805233357577585111418683528, −6.70837816087971111233251865946, −6.66352544892860029015835683131, −6.49837795726161038754314142375, −6.16611103227721296195916685166, −6.09118537405788621281013089458, −6.01683665928884551687878272564, −5.55880597242699760921462607712, −5.52083872024425997673813104689, −5.13729643570304754357811602424, −4.96310940970327705418246472101, −4.53352003046742537680602568600, −4.49499505462255821884902086408, −4.38529663055047805334252454283, −4.28167444796363657382045257402, −3.49025780032061954018839079916, −3.34491376570338424905121438394, −2.85456487422485691404274893742, −2.48777114432843828978640099774, −2.33572953309973904024194901449, −2.33040060687394007776341711173, −2.22855267312832248037897920716, −1.75586070676143837343040325415, −1.19643705564035640918987382617, −1.12183159499626468381880677725, 1.12183159499626468381880677725, 1.19643705564035640918987382617, 1.75586070676143837343040325415, 2.22855267312832248037897920716, 2.33040060687394007776341711173, 2.33572953309973904024194901449, 2.48777114432843828978640099774, 2.85456487422485691404274893742, 3.34491376570338424905121438394, 3.49025780032061954018839079916, 4.28167444796363657382045257402, 4.38529663055047805334252454283, 4.49499505462255821884902086408, 4.53352003046742537680602568600, 4.96310940970327705418246472101, 5.13729643570304754357811602424, 5.52083872024425997673813104689, 5.55880597242699760921462607712, 6.01683665928884551687878272564, 6.09118537405788621281013089458, 6.16611103227721296195916685166, 6.49837795726161038754314142375, 6.66352544892860029015835683131, 6.70837816087971111233251865946, 6.82805233357577585111418683528

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