Properties

Label 8-799e4-1.1-c0e4-0-2
Degree $8$
Conductor $407555836801$
Sign $1$
Analytic cond. $0.0252822$
Root an. cond. $0.631468$
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  + 4·3-s + 10·9-s − 2·16-s − 4·17-s + 20·27-s − 8·48-s − 2·49-s − 16·51-s − 4·53-s − 4·61-s + 4·71-s + 4·79-s + 34·81-s − 4·83-s + 127-s + 131-s + 137-s + 139-s − 20·144-s − 8·147-s + 149-s + 151-s − 40·153-s + 157-s − 16·159-s + 163-s + 167-s + ⋯
L(s)  = 1  + 4·3-s + 10·9-s − 2·16-s − 4·17-s + 20·27-s − 8·48-s − 2·49-s − 16·51-s − 4·53-s − 4·61-s + 4·71-s + 4·79-s + 34·81-s − 4·83-s + 127-s + 131-s + 137-s + 139-s − 20·144-s − 8·147-s + 149-s + 151-s − 40·153-s + 157-s − 16·159-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(17^{4} \cdot 47^{4}\)
Sign: $1$
Analytic conductor: \(0.0252822\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{799} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 17^{4} \cdot 47^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.986613724\)
\(L(\frac12)\) \(\approx\) \(2.986613724\)
\(L(1)\) 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)$
bad17$C_1$ \( ( 1 + T )^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + T^{4} )^{2} \)
3$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
5$C_4\times C_2$ \( 1 + T^{8} \)
7$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_4\times C_2$ \( 1 + T^{8} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
41$C_4\times C_2$ \( 1 + T^{8} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
73$C_4\times C_2$ \( 1 + T^{8} \)
79$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
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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

−7.76807304728872810474613486327, −7.37889227873632408428905342381, −7.32592420002085790698621608933, −6.81267419863250867388740139920, −6.80543112905942585348483167052, −6.72170602454147341278508062384, −6.42239838291363031223512022359, −6.23078548815443055948851362510, −6.08019481742297483042924822901, −5.08739090476776641076545251497, −4.91331005623260117295370210204, −4.79876274307482899583782422644, −4.56935911595939851380123199415, −4.46965917503732683491478775816, −4.11512688321308378648967089916, −3.94593363704164899158115402591, −3.57063600973940416759050567102, −3.35970391375422249321562918347, −2.90150301558418022506540329796, −2.81681354362779972551415009849, −2.49529307446799026054557928504, −2.27441389998862881456962322526, −1.84847062897443005881527195053, −1.74253400638481136706800819564, −1.52141779790946051256389488615, 1.52141779790946051256389488615, 1.74253400638481136706800819564, 1.84847062897443005881527195053, 2.27441389998862881456962322526, 2.49529307446799026054557928504, 2.81681354362779972551415009849, 2.90150301558418022506540329796, 3.35970391375422249321562918347, 3.57063600973940416759050567102, 3.94593363704164899158115402591, 4.11512688321308378648967089916, 4.46965917503732683491478775816, 4.56935911595939851380123199415, 4.79876274307482899583782422644, 4.91331005623260117295370210204, 5.08739090476776641076545251497, 6.08019481742297483042924822901, 6.23078548815443055948851362510, 6.42239838291363031223512022359, 6.72170602454147341278508062384, 6.80543112905942585348483167052, 6.81267419863250867388740139920, 7.32592420002085790698621608933, 7.37889227873632408428905342381, 7.76807304728872810474613486327

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