Properties

Degree $8$
Conductor $4032758016$
Sign $1$
Motivic weight $1$
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 + 6·5-s + 4·8-s + 12·10-s + 8·16-s + 6·17-s + 12·20-s + 11·25-s − 16·29-s + 8·32-s + 12·34-s − 6·37-s + 24·40-s − 2·49-s + 22·50-s − 2·53-s − 32·58-s − 18·61-s + 8·64-s + 12·68-s + 30·73-s − 12·74-s + 48·80-s + 36·85-s − 54·89-s − 4·98-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 2.68·5-s + 1.41·8-s + 3.79·10-s + 2·16-s + 1.45·17-s + 2.68·20-s + 11/5·25-s − 2.97·29-s + 1.41·32-s + 2.05·34-s − 0.986·37-s + 3.79·40-s − 2/7·49-s + 3.11·50-s − 0.274·53-s − 4.20·58-s − 2.30·61-s + 64-s + 1.45·68-s + 3.51·73-s − 1.39·74-s + 5.36·80-s + 3.90·85-s − 5.72·89-s − 0.404·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.25065\)
\(L(\frac12)\) \(\approx\) \(7.25065\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3 \( 1 \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
good5$C_2^2$ \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \)
23$C_2^3$ \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
31$C_2^2$$\times$$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \)
37$C_2^2$ \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 19 T^{2} - 1848 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^3$ \( 1 + 77 T^{2} - 312 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 27 T + 332 T^{2} + 27 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
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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

−9.153773890876115466387162209172, −8.443681402675250788071282963638, −8.116855091635264218708372857772, −8.001127700191252488468349354212, −7.71833459776014426202270358411, −7.35453885307732788569458248129, −7.10986310140454186452835465604, −6.83928423844414598813248344312, −6.63308788477749874791497911961, −5.95553611381268526399009951696, −5.94276809123589883637803203014, −5.76142353580664621155347976565, −5.65455300907543654002181839811, −5.20090128908411526690049517406, −5.17880467708592840293420794750, −4.52740840318411717384902245114, −4.51762269512795833919696319767, −3.88970669704056830956576049036, −3.52404462442427195861886447252, −3.40830712266276344739602648842, −2.93246459607072519749466339585, −2.28880893650311205538905808938, −1.95087455775970083645575875624, −1.74866543200013856322997759698, −1.32934057357336482862740897306, 1.32934057357336482862740897306, 1.74866543200013856322997759698, 1.95087455775970083645575875624, 2.28880893650311205538905808938, 2.93246459607072519749466339585, 3.40830712266276344739602648842, 3.52404462442427195861886447252, 3.88970669704056830956576049036, 4.51762269512795833919696319767, 4.52740840318411717384902245114, 5.17880467708592840293420794750, 5.20090128908411526690049517406, 5.65455300907543654002181839811, 5.76142353580664621155347976565, 5.94276809123589883637803203014, 5.95553611381268526399009951696, 6.63308788477749874791497911961, 6.83928423844414598813248344312, 7.10986310140454186452835465604, 7.35453885307732788569458248129, 7.71833459776014426202270358411, 8.001127700191252488468349354212, 8.116855091635264218708372857772, 8.443681402675250788071282963638, 9.153773890876115466387162209172

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