Properties

Label 8-1344e4-1.1-c1e4-0-21
Degree $8$
Conductor $3.263\times 10^{12}$
Sign $1$
Analytic cond. $13264.9$
Root an. cond. $3.27595$
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  + 4·3-s + 2·7-s + 10·9-s − 12·19-s + 8·21-s + 6·25-s + 20·27-s + 8·29-s + 12·37-s − 8·47-s + 6·49-s − 16·53-s − 48·57-s − 16·59-s + 20·63-s + 24·75-s + 35·81-s + 8·83-s + 32·87-s − 32·103-s + 48·111-s + 16·113-s + 34·121-s + 127-s + 131-s − 24·133-s + 137-s + ⋯
L(s)  = 1  + 2.30·3-s + 0.755·7-s + 10/3·9-s − 2.75·19-s + 1.74·21-s + 6/5·25-s + 3.84·27-s + 1.48·29-s + 1.97·37-s − 1.16·47-s + 6/7·49-s − 2.19·53-s − 6.35·57-s − 2.08·59-s + 2.51·63-s + 2.77·75-s + 35/9·81-s + 0.878·83-s + 3.43·87-s − 3.15·103-s + 4.55·111-s + 1.50·113-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(13264.9\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.677406467\)
\(L(\frac12)\) \(\approx\) \(9.677406467\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2 \wr C_2$ \( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 - 22 T^{2} + 682 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 82 T^{2} + 2722 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$D_{4}$ \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - 102 T^{2} + 5130 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2 \wr C_2$ \( 1 - 24 T^{2} + 3774 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
61$C_2^2 \wr C_2$ \( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2 \wr C_2$ \( 1 - 144 T^{2} + 10830 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 - 114 T^{2} + 8418 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2 \wr C_2$ \( 1 - 236 T^{2} + 24310 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2 \wr C_2$ \( 1 - 288 T^{2} + 33150 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2 \wr C_2$ \( 1 - 294 T^{2} + 36618 T^{4} - 294 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.89036646959552419444126580089, −6.50939058862327031799228643818, −6.44388256344380015865857769981, −6.35491217561144620350206632060, −6.28696026969846933133812442484, −5.87561353580436417830888764364, −5.37927404813374865812190658671, −5.18906199346425274035197626012, −5.07410898654795042365793588850, −4.48123100252650329653451117933, −4.48086974115770358996546663894, −4.46726537021177509265900310762, −4.24333295336788032163965425361, −3.80569001824007181044476527272, −3.74491467132211435780822346536, −3.05273972029844201553494642907, −3.01938622932176995439183910138, −2.88910444178870556983193640805, −2.77323960001762278070916731112, −2.11838986608880258190419627611, −2.02561978360201775546841953808, −1.83274206804510419991550996831, −1.44197940744626500038894313153, −1.03631049407279061572308016740, −0.48262282007089484425203160717, 0.48262282007089484425203160717, 1.03631049407279061572308016740, 1.44197940744626500038894313153, 1.83274206804510419991550996831, 2.02561978360201775546841953808, 2.11838986608880258190419627611, 2.77323960001762278070916731112, 2.88910444178870556983193640805, 3.01938622932176995439183910138, 3.05273972029844201553494642907, 3.74491467132211435780822346536, 3.80569001824007181044476527272, 4.24333295336788032163965425361, 4.46726537021177509265900310762, 4.48086974115770358996546663894, 4.48123100252650329653451117933, 5.07410898654795042365793588850, 5.18906199346425274035197626012, 5.37927404813374865812190658671, 5.87561353580436417830888764364, 6.28696026969846933133812442484, 6.35491217561144620350206632060, 6.44388256344380015865857769981, 6.50939058862327031799228643818, 6.89036646959552419444126580089

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