Properties

Label 8-1620e4-1.1-c0e4-0-5
Degree $8$
Conductor $6.887\times 10^{12}$
Sign $1$
Analytic cond. $0.427256$
Root an. cond. $0.899158$
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  + 4-s + 25-s + 2·49-s + 4·61-s − 64-s + 100-s + 8·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 4-s + 25-s + 2·49-s + 4·61-s − 64-s + 100-s + 8·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.427256\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.705725655\)
\(L(\frac12)\) \(\approx\) \(1.705725655\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3 \( 1 \)
5$C_2^2$ \( 1 - T^{2} + T^{4} \)
good7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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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

−7.04057901194451708312205217036, −6.54963811539873896149969850818, −6.54032462287971348434811516737, −6.36349885480488703167579885362, −5.94122735338285999926038035738, −5.87940670171164686388235316514, −5.83014395926782614067774759010, −5.31831640933861909176935274380, −5.17139285300270339867545575216, −4.99548007048139648599869820669, −4.78424812946469498178519331162, −4.39507910584475596076420157303, −4.36669788582558019045778846385, −3.91024108092007094341439498897, −3.76447165701797227382057503573, −3.39895092753292895614691590449, −3.32885454178301273009373594473, −3.04657426948438142028518282849, −2.48602737220232797426061366771, −2.41088543567174479381565954539, −2.23628494430828299140949990625, −2.06023936028566122535163056463, −1.51862118490874178132704767725, −0.975806822276998642897698751818, −0.925882931185934536573689556123, 0.925882931185934536573689556123, 0.975806822276998642897698751818, 1.51862118490874178132704767725, 2.06023936028566122535163056463, 2.23628494430828299140949990625, 2.41088543567174479381565954539, 2.48602737220232797426061366771, 3.04657426948438142028518282849, 3.32885454178301273009373594473, 3.39895092753292895614691590449, 3.76447165701797227382057503573, 3.91024108092007094341439498897, 4.36669788582558019045778846385, 4.39507910584475596076420157303, 4.78424812946469498178519331162, 4.99548007048139648599869820669, 5.17139285300270339867545575216, 5.31831640933861909176935274380, 5.83014395926782614067774759010, 5.87940670171164686388235316514, 5.94122735338285999926038035738, 6.36349885480488703167579885362, 6.54032462287971348434811516737, 6.54963811539873896149969850818, 7.04057901194451708312205217036

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