Properties

Label 8-882e4-1.1-c1e4-0-8
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $2460.26$
Root an. cond. $2.65382$
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  + 2·2-s + 3-s + 4-s − 6·5-s + 2·6-s − 2·8-s + 3·9-s − 12·10-s + 6·11-s + 12-s + 4·13-s − 6·15-s − 4·16-s − 3·17-s + 6·18-s + 10·19-s − 6·20-s + 12·22-s − 18·23-s − 2·24-s + 19·25-s + 8·26-s + 8·27-s + 6·29-s − 12·30-s + 4·31-s − 2·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 1/2·4-s − 2.68·5-s + 0.816·6-s − 0.707·8-s + 9-s − 3.79·10-s + 1.80·11-s + 0.288·12-s + 1.10·13-s − 1.54·15-s − 16-s − 0.727·17-s + 1.41·18-s + 2.29·19-s − 1.34·20-s + 2.55·22-s − 3.75·23-s − 0.408·24-s + 19/5·25-s + 1.56·26-s + 1.53·27-s + 1.11·29-s − 2.19·30-s + 0.718·31-s − 0.353·32-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2460.26\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.640694872\)
\(L(\frac12)\) \(\approx\) \(5.640694872\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
17$D_4\times C_2$ \( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 720 p T^{5} + 95 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 6 T - 46 T^{2} + 144 T^{3} + 2007 T^{4} + 144 p T^{5} - 46 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 3 T - 37 T^{2} - 216 T^{3} - 1896 T^{4} - 216 p T^{5} - 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 11 T + 43 T^{2} - 484 T^{3} - 5018 T^{4} - 484 p T^{5} + 43 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} ) \)
71$D_{4}$ \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 7 T - 35 T^{2} + 434 T^{3} - 1850 T^{4} + 434 p T^{5} - 35 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 1152 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 18 T + 98 T^{2} - 864 T^{3} + 14319 T^{4} - 864 p T^{5} + 98 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - T - 119 T^{2} + 74 T^{3} + 4894 T^{4} + 74 p T^{5} - 119 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.52368495856780797072472584769, −6.86956326168829602931915229316, −6.81539802458336483448266940896, −6.62134394489382026616208348475, −6.34926936412669383505700528566, −6.14257766890731202948227007977, −5.89954786729556636901363134448, −5.71541173447312992321421994251, −5.37155634502843482514978187954, −4.82306960102525839619598381578, −4.74085621705536424090567824829, −4.66514172108095249548576105838, −4.11533252274311977013362676852, −4.07863961872967698723859359383, −4.06701900198906911448336074545, −3.70229909207548373804336866655, −3.68067271713760049751381038426, −3.18625599602675634945660049184, −2.99243127385315125048316031527, −2.80853770388988840838640433604, −2.23625729788867751514618077490, −1.73407927815907374175459666774, −1.45368049449275543607068222721, −0.77812527821238270647566032634, −0.62027178633239084591820805985, 0.62027178633239084591820805985, 0.77812527821238270647566032634, 1.45368049449275543607068222721, 1.73407927815907374175459666774, 2.23625729788867751514618077490, 2.80853770388988840838640433604, 2.99243127385315125048316031527, 3.18625599602675634945660049184, 3.68067271713760049751381038426, 3.70229909207548373804336866655, 4.06701900198906911448336074545, 4.07863961872967698723859359383, 4.11533252274311977013362676852, 4.66514172108095249548576105838, 4.74085621705536424090567824829, 4.82306960102525839619598381578, 5.37155634502843482514978187954, 5.71541173447312992321421994251, 5.89954786729556636901363134448, 6.14257766890731202948227007977, 6.34926936412669383505700528566, 6.62134394489382026616208348475, 6.81539802458336483448266940896, 6.86956326168829602931915229316, 7.52368495856780797072472584769

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