Properties

Label 8-4608e4-1.1-c1e4-0-8
Degree $8$
Conductor $4.509\times 10^{14}$
Sign $1$
Analytic cond. $1.83298\times 10^{6}$
Root an. cond. $6.06589$
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  + 8·11-s − 8·17-s − 24·41-s + 16·43-s + 16·59-s + 32·67-s + 16·73-s + 24·83-s − 40·89-s + 24·97-s − 16·107-s − 24·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s − 64·187-s + ⋯
L(s)  = 1  + 2.41·11-s − 1.94·17-s − 3.74·41-s + 2.43·43-s + 2.08·59-s + 3.90·67-s + 1.87·73-s + 2.63·83-s − 4.23·89-s + 2.43·97-s − 1.54·107-s − 2.25·113-s + 1.09·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 4.68·187-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.83298\times 10^{6}\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.123099769\)
\(L(\frac12)\) \(\approx\) \(2.123099769\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$D_4\times C_2$ \( 1 + 18 T^{4} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 30 T^{4} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2 \wr C_2$ \( 1 + 12 T^{2} + 246 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 + 12 T^{2} + 582 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 1650 T^{4} + p^{4} T^{8} \)
31$C_2^2 \wr C_2$ \( 1 + 96 T^{2} + 4098 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2 \wr C_2$ \( 1 + 92 T^{2} + 4342 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 + 108 T^{2} + 6822 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 + 64 T^{2} + 1234 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2 \wr C_2$ \( 1 + 188 T^{2} + 15766 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 + 172 T^{2} + 15430 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
79$C_2^2 \wr C_2$ \( 1 + 288 T^{2} + 33090 T^{4} + 288 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
97$D_{4}$ \( ( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.10486751256755172073102709834, −5.48368263522844979385062994312, −5.39101405367488338520510000972, −5.29447791588718935174009823266, −5.24580544107918271994435847522, −4.86241744375696615736814355207, −4.70627976007212790487739223705, −4.35276585406584148815087713272, −4.31003070686305185061309963527, −3.98700225523211816828518473187, −3.84190925617764181753777453587, −3.76412311754615987972267465554, −3.64070115164255445999272285527, −3.32731724529690228650556818709, −2.96420341804041247937187789675, −2.76155388841597234388065322812, −2.57002486619137647485707925793, −2.14457032456994404359807114212, −1.96044355940959944336558496052, −1.93600191184547583681500063077, −1.68524460678040110668288651810, −1.06480502007850324764970031627, −0.980088908828635200669023482698, −0.78036567888018742138106023790, −0.19068813614980585084005642779, 0.19068813614980585084005642779, 0.78036567888018742138106023790, 0.980088908828635200669023482698, 1.06480502007850324764970031627, 1.68524460678040110668288651810, 1.93600191184547583681500063077, 1.96044355940959944336558496052, 2.14457032456994404359807114212, 2.57002486619137647485707925793, 2.76155388841597234388065322812, 2.96420341804041247937187789675, 3.32731724529690228650556818709, 3.64070115164255445999272285527, 3.76412311754615987972267465554, 3.84190925617764181753777453587, 3.98700225523211816828518473187, 4.31003070686305185061309963527, 4.35276585406584148815087713272, 4.70627976007212790487739223705, 4.86241744375696615736814355207, 5.24580544107918271994435847522, 5.29447791588718935174009823266, 5.39101405367488338520510000972, 5.48368263522844979385062994312, 6.10486751256755172073102709834

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