Properties

Label 8-1200e4-1.1-c2e4-0-13
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·9-s + 8·19-s + 88·31-s + 124·49-s − 344·61-s + 40·79-s + 115·81-s − 40·109-s + 420·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s + 112·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 14/9·9-s + 8/19·19-s + 2.83·31-s + 2.53·49-s − 5.63·61-s + 0.506·79-s + 1.41·81-s − 0.366·109-s + 3.47·121-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 + 2.81·169-s + 0.654·171-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

\(L(\frac{3}{2})\) \(\approx\) \(1.665025977\)
\(L(\frac12)\) \(\approx\) \(1.665025977\)
\(L(2)\) 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$C_2^2$ \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 210 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + 66 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 930 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1394 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 2702 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2210 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 3026 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 1746 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 1554 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 86 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 8974 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 5406 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 3934 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 8370 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 14690 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 9982 T^{2} + p^{4} T^{4} )^{2} \)
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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

−6.60913477039855778845591896451, −6.56768566136938710418945860978, −6.37114532820351870517863009458, −6.18654984821441624099324595274, −5.97473898376492286837498999175, −5.66343047953235140065319952628, −5.41632925145245739796873318481, −5.05577471395868185239380443949, −5.03814207501648209222907801610, −4.57185036419640137831163729311, −4.40779874694808287284868531541, −4.37888765129201213241982728690, −4.03068789230044918146839830424, −3.94759266142029296224158121254, −3.27662824567679045423743814768, −3.18881363189275794261098118281, −3.07566771995272463763786919740, −2.69425762212409469461542112496, −2.30926691989195602318665824048, −2.02869926988907983939862029892, −1.78914243077463375638658594177, −1.20575324957054945883953449563, −1.05465217151101387203679092698, −0.947275998151289551516037014993, −0.16456326129003196006968900426, 0.16456326129003196006968900426, 0.947275998151289551516037014993, 1.05465217151101387203679092698, 1.20575324957054945883953449563, 1.78914243077463375638658594177, 2.02869926988907983939862029892, 2.30926691989195602318665824048, 2.69425762212409469461542112496, 3.07566771995272463763786919740, 3.18881363189275794261098118281, 3.27662824567679045423743814768, 3.94759266142029296224158121254, 4.03068789230044918146839830424, 4.37888765129201213241982728690, 4.40779874694808287284868531541, 4.57185036419640137831163729311, 5.03814207501648209222907801610, 5.05577471395868185239380443949, 5.41632925145245739796873318481, 5.66343047953235140065319952628, 5.97473898376492286837498999175, 6.18654984821441624099324595274, 6.37114532820351870517863009458, 6.56768566136938710418945860978, 6.60913477039855778845591896451

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