Properties

Label 8-1200e4-1.1-c1e4-0-3
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s + 24·17-s − 4·49-s − 24·53-s − 40·61-s + 27·81-s − 40·109-s + 24·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 144·153-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·9-s + 5.82·17-s − 4/7·49-s − 3.29·53-s − 5.12·61-s + 3·81-s − 3.83·109-s + 2.25·113-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 − 11.6·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-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 + ⋯

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: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4281722287\)
\(L(\frac12)\) \(\approx\) \(0.4281722287\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.96153231702213991202372303889, −6.63064684602502409705402700515, −6.33512694048389246138360723196, −6.30500233653912990937573092000, −6.07309799562132128577814309949, −5.82147497366625092112464639473, −5.61958922686158489587338249905, −5.35291526840958205757561686416, −5.25520674165005389280692451914, −5.21391367531390661488275909399, −4.74217799323995507355433256998, −4.52109332751503868758612936892, −4.26529821317504420513894404820, −3.77005920092630376544209557033, −3.44123645151480479574393018605, −3.33557797184952633163442298521, −3.26070601934109285016083911164, −3.02466625771428007346740896713, −2.76202907071169921014993198063, −2.49288223119239946990374059845, −1.90991962484869984031872363737, −1.35616875253277572459387037971, −1.28130334964416257734969776934, −1.13398123254098601259281455735, −0.13913519217452760597168849125, 0.13913519217452760597168849125, 1.13398123254098601259281455735, 1.28130334964416257734969776934, 1.35616875253277572459387037971, 1.90991962484869984031872363737, 2.49288223119239946990374059845, 2.76202907071169921014993198063, 3.02466625771428007346740896713, 3.26070601934109285016083911164, 3.33557797184952633163442298521, 3.44123645151480479574393018605, 3.77005920092630376544209557033, 4.26529821317504420513894404820, 4.52109332751503868758612936892, 4.74217799323995507355433256998, 5.21391367531390661488275909399, 5.25520674165005389280692451914, 5.35291526840958205757561686416, 5.61958922686158489587338249905, 5.82147497366625092112464639473, 6.07309799562132128577814309949, 6.30500233653912990937573092000, 6.33512694048389246138360723196, 6.63064684602502409705402700515, 6.96153231702213991202372303889

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