Properties

Label 8-1400e4-1.1-c0e4-0-0
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $0.238309$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·11-s − 16-s − 2·81-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 8·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 8·11-s − 16-s − 2·81-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 8·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.238309\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03073970958\)
\(L(\frac12)\) \(\approx\) \(0.03073970958\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good3$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_1$ \( ( 1 + T )^{8} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−7.10921833043693498197934647900, −6.98155645399034343830341788515, −6.76176677786536511397659401339, −6.29603218823473663285563759050, −5.87232760643823218175924248374, −5.83622774236759075878055463790, −5.63856108595301854328853032195, −5.58153189920996951831096611453, −5.22282530444739120148876087949, −5.05297713927023676042057602166, −4.89851286373359191943356363649, −4.70745239620079971888094404548, −4.55386295984599860582520535068, −4.32983848716210785356525031360, −3.79422946490486490336224424486, −3.36824504198896216897302967310, −3.19189603220687902237035375870, −2.86991368829625827495916447409, −2.84433119975387098115821288714, −2.43216942156901137481886955249, −2.36079290218657648514300410497, −2.15372310593485471066463821046, −1.88065177031135032675048080936, −0.990213845625907065929441093178, −0.11420583153034822089423797756, 0.11420583153034822089423797756, 0.990213845625907065929441093178, 1.88065177031135032675048080936, 2.15372310593485471066463821046, 2.36079290218657648514300410497, 2.43216942156901137481886955249, 2.84433119975387098115821288714, 2.86991368829625827495916447409, 3.19189603220687902237035375870, 3.36824504198896216897302967310, 3.79422946490486490336224424486, 4.32983848716210785356525031360, 4.55386295984599860582520535068, 4.70745239620079971888094404548, 4.89851286373359191943356363649, 5.05297713927023676042057602166, 5.22282530444739120148876087949, 5.58153189920996951831096611453, 5.63856108595301854328853032195, 5.83622774236759075878055463790, 5.87232760643823218175924248374, 6.29603218823473663285563759050, 6.76176677786536511397659401339, 6.98155645399034343830341788515, 7.10921833043693498197934647900

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