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Properties

Label 8-1792e4-1.1-c0e4-0-1
Degree $8$
Conductor $1.031\times 10^{13}$
Sign $1$
Analytic cond. $0.639706$
Root an. cond. $0.945687$
Motivic weight $0$
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  − 4·23-s + 4·43-s + 4·53-s − 4·67-s − 4·107-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 4·23-s + 4·43-s + 4·53-s − 4·67-s − 4·107-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.639706\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8077651005\)
\(L(\frac12)\) \(\approx\) \(0.8077651005\)
\(L(1)\) 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 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.74348346996390722463628708819, −6.49082120131278906716560222498, −6.30294304954170728983797977423, −6.13333274453785129025115827209, −6.12566235568468787622070592799, −5.64521292466912601895852583519, −5.48419289521583337998148257676, −5.43910631144320791523523074972, −5.33239588157186812095336307573, −4.90697196190612122409324870905, −4.35406246107934631704726748596, −4.34433829279644104583674958906, −4.16867116318732167491903933302, −3.86133822578618242995045437430, −3.85460609620413376151389506013, −3.83404111030197121851626879394, −3.06884275205485604955753483756, −2.83770628128964537346488959231, −2.57831894953406648589561554022, −2.40960796835518361618559302744, −2.32250155642156792449899178178, −1.74648958610442668539868404514, −1.41513987497040380049030504688, −1.28299193648057815971937940009, −0.48535134911256242876957148613, 0.48535134911256242876957148613, 1.28299193648057815971937940009, 1.41513987497040380049030504688, 1.74648958610442668539868404514, 2.32250155642156792449899178178, 2.40960796835518361618559302744, 2.57831894953406648589561554022, 2.83770628128964537346488959231, 3.06884275205485604955753483756, 3.83404111030197121851626879394, 3.85460609620413376151389506013, 3.86133822578618242995045437430, 4.16867116318732167491903933302, 4.34433829279644104583674958906, 4.35406246107934631704726748596, 4.90697196190612122409324870905, 5.33239588157186812095336307573, 5.43910631144320791523523074972, 5.48419289521583337998148257676, 5.64521292466912601895852583519, 6.12566235568468787622070592799, 6.13333274453785129025115827209, 6.30294304954170728983797977423, 6.49082120131278906716560222498, 6.74348346996390722463628708819

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