Properties

Label 8-1792e4-1.1-c0e4-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.390629537\)
\(L(\frac12)\) \(\approx\) \(1.390629537\)
\(L(1)\) 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

−6.94275584401190643759361144731, −6.51609717971453323263356688353, −6.45542394977899089445833295075, −6.23276962458764072875551943722, −6.21576793495913288672666938555, −5.46685489050827273220398100456, −5.37455811743892281392160070934, −5.25395544587937548126499269955, −5.24304145565766964589301061534, −4.93392771874520524762208233616, −4.82380898619976756580846364012, −4.47813848181856758167666029695, −4.16009018173881436333901245907, −3.71071883275013071287590434628, −3.70794689649359819933394860728, −3.50345252333540068860612328111, −3.40043955131161434587863848562, −2.75221736580233919425873058706, −2.69214100945312238063006563296, −2.42588627887330290458188613574, −2.35015299267904367884262980792, −1.70070567771751940877389402323, −1.30402428085307831129897323018, −1.21016810437083424934966732154, −0.72326751539914974162141785262, 0.72326751539914974162141785262, 1.21016810437083424934966732154, 1.30402428085307831129897323018, 1.70070567771751940877389402323, 2.35015299267904367884262980792, 2.42588627887330290458188613574, 2.69214100945312238063006563296, 2.75221736580233919425873058706, 3.40043955131161434587863848562, 3.50345252333540068860612328111, 3.70794689649359819933394860728, 3.71071883275013071287590434628, 4.16009018173881436333901245907, 4.47813848181856758167666029695, 4.82380898619976756580846364012, 4.93392771874520524762208233616, 5.24304145565766964589301061534, 5.25395544587937548126499269955, 5.37455811743892281392160070934, 5.46685489050827273220398100456, 6.21576793495913288672666938555, 6.23276962458764072875551943722, 6.45542394977899089445833295075, 6.51609717971453323263356688353, 6.94275584401190643759361144731

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