Properties

Label 8-1200e4-1.1-c2e4-0-10
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $1.14304\times 10^{6}$
Root an. cond. $5.71818$
Motivic weight $2$
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  + 14·9-s − 28·19-s + 28·31-s + 98·49-s − 4·61-s − 232·79-s + 115·81-s + 100·109-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 574·169-s − 392·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 14/9·9-s − 1.47·19-s + 0.903·31-s + 2·49-s − 0.0655·61-s − 2.93·79-s + 1.41·81-s + 0.917·109-s + 2.80·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.39·169-s − 2.29·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.14304\times 10^{6}\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.320271042\)
\(L(\frac12)\) \(\approx\) \(1.320271042\)
\(L(2)\) 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$C_2^2$ \( 1 - 14 T^{2} + p^{4} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 287 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 70 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 410 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 118 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 2734 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 3290 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2017 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 2090 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 5810 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 8689 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 8282 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 5758 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 58 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 2590 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 16417 T^{2} + p^{4} T^{4} )^{2} \)
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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

−6.64278964451902735253415329976, −6.60181790402161353176180922733, −6.40411264189723278383452874385, −6.06019561065432534927676230370, −5.97069480624930357531892700891, −5.65011818765130617805514609886, −5.55600006775093966164655338703, −5.03854970225110971379856822461, −4.93108165515198497351579105964, −4.71503810520521658471173896153, −4.36684393713173736132082266467, −4.27732212041454734113724923231, −4.02391856487902037429145933374, −3.94755524780245079122928896907, −3.36327907802814100333814439448, −3.33440985171455197597220348942, −2.99214501888741818460519181355, −2.43831101400054183109312140101, −2.34940723502259828140793591411, −2.19666061526105055707763400553, −1.68771663962091490547487821082, −1.37836683145738001812477155565, −1.09540097078579645140606667992, −0.76144653249369402782094852072, −0.15869151030598197081948981726, 0.15869151030598197081948981726, 0.76144653249369402782094852072, 1.09540097078579645140606667992, 1.37836683145738001812477155565, 1.68771663962091490547487821082, 2.19666061526105055707763400553, 2.34940723502259828140793591411, 2.43831101400054183109312140101, 2.99214501888741818460519181355, 3.33440985171455197597220348942, 3.36327907802814100333814439448, 3.94755524780245079122928896907, 4.02391856487902037429145933374, 4.27732212041454734113724923231, 4.36684393713173736132082266467, 4.71503810520521658471173896153, 4.93108165515198497351579105964, 5.03854970225110971379856822461, 5.55600006775093966164655338703, 5.65011818765130617805514609886, 5.97069480624930357531892700891, 6.06019561065432534927676230370, 6.40411264189723278383452874385, 6.60181790402161353176180922733, 6.64278964451902735253415329976

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