Properties

Label 8-28e8-1.1-c1e4-0-2
Degree $8$
Conductor $377801998336$
Sign $1$
Analytic cond. $1535.93$
Root an. cond. $2.50205$
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 − 4·3-s + 2·4-s − 4·5-s − 8·6-s + 4·8-s + 8·9-s − 8·10-s + 6·11-s − 8·12-s + 16·15-s + 8·16-s − 12·17-s + 16·18-s − 8·19-s − 8·20-s + 12·22-s − 16·24-s + 8·25-s − 8·27-s − 4·29-s + 32·30-s + 8·31-s + 8·32-s − 24·33-s − 24·34-s + 16·36-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s + 4-s − 1.78·5-s − 3.26·6-s + 1.41·8-s + 8/3·9-s − 2.52·10-s + 1.80·11-s − 2.30·12-s + 4.13·15-s + 2·16-s − 2.91·17-s + 3.77·18-s − 1.83·19-s − 1.78·20-s + 2.55·22-s − 3.26·24-s + 8/5·25-s − 1.53·27-s − 0.742·29-s + 5.84·30-s + 1.43·31-s + 1.41·32-s − 4.17·33-s − 4.11·34-s + 8/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1535.93\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7069831063\)
\(L(\frac12)\) \(\approx\) \(0.7069831063\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good3$C_2^3$ \( 1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2^3$ \( 1 + 4 T + 8 T^{2} - 8 T^{3} - 41 T^{4} - 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^3$ \( 1 - 6 T + 18 T^{2} + 24 T^{3} - 193 T^{4} + 24 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^3$ \( 1 + 8 T + 32 T^{2} - 48 T^{3} - 553 T^{4} - 48 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^3$ \( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 336 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 + 14 T + 98 T^{2} - 112 T^{3} - 3593 T^{4} - 112 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 4 T + 8 T^{2} + 440 T^{3} - 4361 T^{4} + 440 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 12 T + 72 T^{2} + 600 T^{3} - 7321 T^{4} + 600 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 10 T + 50 T^{2} + 840 T^{3} - 8689 T^{4} + 840 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 142 T^{2} + 12243 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
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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

−6.95299222741407454729971303404, −6.93892861065356098741770633174, −6.75276900931045612323096726561, −6.67518953996557328205203395397, −6.66756189447849204998684615178, −6.17378493065671124081982651026, −6.09396410369004684736591324105, −5.79426508007295820960791735620, −5.38815780835485413125841473125, −5.19428854032495937602208970643, −4.80921689755316894843614613934, −4.60806475060633600469588354625, −4.56399133983903816277211614079, −4.53840031563731185723751945666, −4.23050349544718502931429607685, −3.76687787982593349307910996315, −3.54384390167329254030708209725, −3.48264410196605567004355467369, −3.20573548775911281633349411067, −2.44061151621774056076985232118, −1.98251858720647268035320530976, −1.88419214746094262672510623099, −1.47578326776615152773069707485, −0.71599611012821947614742845731, −0.28576856576474805902487597112, 0.28576856576474805902487597112, 0.71599611012821947614742845731, 1.47578326776615152773069707485, 1.88419214746094262672510623099, 1.98251858720647268035320530976, 2.44061151621774056076985232118, 3.20573548775911281633349411067, 3.48264410196605567004355467369, 3.54384390167329254030708209725, 3.76687787982593349307910996315, 4.23050349544718502931429607685, 4.53840031563731185723751945666, 4.56399133983903816277211614079, 4.60806475060633600469588354625, 4.80921689755316894843614613934, 5.19428854032495937602208970643, 5.38815780835485413125841473125, 5.79426508007295820960791735620, 6.09396410369004684736591324105, 6.17378493065671124081982651026, 6.66756189447849204998684615178, 6.67518953996557328205203395397, 6.75276900931045612323096726561, 6.93892861065356098741770633174, 6.95299222741407454729971303404

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