Properties

Label 8-672e4-1.1-c1e4-0-1
Degree $8$
Conductor $203928109056$
Sign $1$
Analytic cond. $829.059$
Root an. cond. $2.31645$
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  + 2·3-s + 4·5-s + 2·9-s + 8·15-s − 20·19-s − 16·23-s + 6·27-s − 16·43-s + 8·45-s + 8·47-s − 2·49-s − 24·53-s − 40·57-s + 8·67-s − 32·69-s − 8·71-s + 16·73-s + 11·81-s − 80·95-s + 32·97-s − 52·101-s − 64·115-s + 4·121-s − 20·125-s + 127-s − 32·129-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 2/3·9-s + 2.06·15-s − 4.58·19-s − 3.33·23-s + 1.15·27-s − 2.43·43-s + 1.19·45-s + 1.16·47-s − 2/7·49-s − 3.29·53-s − 5.29·57-s + 0.977·67-s − 3.85·69-s − 0.949·71-s + 1.87·73-s + 11/9·81-s − 8.20·95-s + 3.24·97-s − 5.17·101-s − 5.96·115-s + 4/11·121-s − 1.78·125-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(829.059\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5792475964\)
\(L(\frac12)\) \(\approx\) \(0.5792475964\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
19$D_{4}$ \( ( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
43$D_{4}$ \( ( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 224 T^{2} + 19486 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 104 T^{2} + 5646 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_4$ \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 124 T^{2} + 11206 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 80 T^{2} + 12958 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.70321722261000367576773512329, −7.34887582644945276889519399589, −7.10051262179613694444387755747, −6.59919948266156650455513040904, −6.48655236715676557352128848136, −6.34919369670610340982207952401, −6.26274766130226540521493806937, −6.12903207696050279927386826633, −5.70671892910726035786767975308, −5.54088473557920677750225394549, −5.06075724820544981892478111924, −4.92841506400247121958072650782, −4.42199383630428046038802185386, −4.41155465914585970432949592283, −4.08642522780298187312639876311, −3.77760617853955280839338698777, −3.59011774973407130960702526833, −3.25660599623607248556918686877, −2.55494884535674399791079643881, −2.50391266319165220192892077472, −2.28012175826183377934055969070, −1.85058141211833865736529027809, −1.76525964542637424627347857019, −1.54972273034918846143532238428, −0.15941993248579120173184161795, 0.15941993248579120173184161795, 1.54972273034918846143532238428, 1.76525964542637424627347857019, 1.85058141211833865736529027809, 2.28012175826183377934055969070, 2.50391266319165220192892077472, 2.55494884535674399791079643881, 3.25660599623607248556918686877, 3.59011774973407130960702526833, 3.77760617853955280839338698777, 4.08642522780298187312639876311, 4.41155465914585970432949592283, 4.42199383630428046038802185386, 4.92841506400247121958072650782, 5.06075724820544981892478111924, 5.54088473557920677750225394549, 5.70671892910726035786767975308, 6.12903207696050279927386826633, 6.26274766130226540521493806937, 6.34919369670610340982207952401, 6.48655236715676557352128848136, 6.59919948266156650455513040904, 7.10051262179613694444387755747, 7.34887582644945276889519399589, 7.70321722261000367576773512329

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