Properties

Label 8-260e4-1.1-c1e4-0-1
Degree $8$
Conductor $4569760000$
Sign $1$
Analytic cond. $18.5781$
Root an. cond. $1.44087$
Motivic weight $1$
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  + 2·2-s + 2·4-s − 4·5-s + 4·8-s − 8·10-s − 6·13-s + 8·16-s − 4·17-s − 8·20-s + 5·25-s − 12·26-s − 30·29-s + 8·32-s − 8·34-s − 22·37-s − 16·40-s + 24·41-s + 10·50-s − 12·52-s − 10·53-s − 60·58-s − 12·61-s + 8·64-s + 24·65-s − 8·68-s + 22·73-s − 44·74-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.78·5-s + 1.41·8-s − 2.52·10-s − 1.66·13-s + 2·16-s − 0.970·17-s − 1.78·20-s + 25-s − 2.35·26-s − 5.57·29-s + 1.41·32-s − 1.37·34-s − 3.61·37-s − 2.52·40-s + 3.74·41-s + 1.41·50-s − 1.66·52-s − 1.37·53-s − 7.87·58-s − 1.53·61-s + 64-s + 2.97·65-s − 0.970·68-s + 2.57·73-s − 5.11·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(18.5781\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{260} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.300859861\)
\(L(\frac12)\) \(\approx\) \(1.300859861\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
good3$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
7$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$$\times$$C_2^2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \)
41$C_2$$\times$$C_2^2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \)
43$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
59$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$$\times$$C_2^2$ \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \)
67$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \)
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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

−8.792035243466649634966189173722, −8.400247076931231053490619797675, −7.87783439236222089542648807109, −7.79550374872632366999366220395, −7.60878650336458469676493595936, −7.55302108494919958138867422268, −7.11521259470731472799974629183, −7.01104468523068032294358559032, −6.91142190369548993287531999010, −6.12655603399409297033607756076, −5.89933104727720344581735926307, −5.67732661208303335685770569564, −5.56958272668486288646026675146, −4.86561018704451972422094923930, −4.80979600130612078434710185279, −4.49988109252360874053445407454, −4.49210578042835974567409388455, −3.76036702768594012231891319056, −3.53397698297726739596700015736, −3.46000644685835524707585614927, −3.37800539749564835381506697996, −2.13845651884361027847611206064, −2.08898932961594158668462306208, −1.95942638812661455191096502919, −0.43759024778853351831377871820, 0.43759024778853351831377871820, 1.95942638812661455191096502919, 2.08898932961594158668462306208, 2.13845651884361027847611206064, 3.37800539749564835381506697996, 3.46000644685835524707585614927, 3.53397698297726739596700015736, 3.76036702768594012231891319056, 4.49210578042835974567409388455, 4.49988109252360874053445407454, 4.80979600130612078434710185279, 4.86561018704451972422094923930, 5.56958272668486288646026675146, 5.67732661208303335685770569564, 5.89933104727720344581735926307, 6.12655603399409297033607756076, 6.91142190369548993287531999010, 7.01104468523068032294358559032, 7.11521259470731472799974629183, 7.55302108494919958138867422268, 7.60878650336458469676493595936, 7.79550374872632366999366220395, 7.87783439236222089542648807109, 8.400247076931231053490619797675, 8.792035243466649634966189173722

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