Properties

Label 8-380e4-1.1-c1e4-0-1
Degree $8$
Conductor $20851360000$
Sign $1$
Analytic cond. $84.7701$
Root an. cond. $1.74192$
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 − 6·3-s + 2·4-s − 2·5-s − 12·6-s + 4·8-s + 18·9-s − 4·10-s − 12·12-s + 2·13-s + 12·15-s + 8·16-s + 6·17-s + 36·18-s − 4·20-s + 12·23-s − 24·24-s + 5·25-s + 4·26-s − 36·27-s + 24·30-s + 8·32-s + 12·34-s + 36·36-s + 8·37-s − 12·39-s − 8·40-s + ⋯
L(s)  = 1  + 1.41·2-s − 3.46·3-s + 4-s − 0.894·5-s − 4.89·6-s + 1.41·8-s + 6·9-s − 1.26·10-s − 3.46·12-s + 0.554·13-s + 3.09·15-s + 2·16-s + 1.45·17-s + 8.48·18-s − 0.894·20-s + 2.50·23-s − 4.89·24-s + 25-s + 0.784·26-s − 6.92·27-s + 4.38·30-s + 1.41·32-s + 2.05·34-s + 6·36-s + 1.31·37-s − 1.92·39-s − 1.26·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(84.7701\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.652791238\)
\(L(\frac12)\) \(\approx\) \(1.652791238\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + p T + p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
7$C_2^3$ \( 1 + 2 T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
23$C_2^3$ \( 1 - 12 T + 72 T^{2} - 288 T^{3} + 1151 T^{4} - 288 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^3$ \( 1 + 57 T^{2} + 2408 T^{4} + 57 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 791 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^3$ \( 1 - 30 T + 450 T^{2} - 4500 T^{3} + 34391 T^{4} - 4500 p T^{5} + 450 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 6 T + 18 T^{2} + 528 T^{3} - 4393 T^{4} + 528 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 6 T + 18 T^{2} + 36 T^{3} - 3649 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 - 16 T + 128 T^{2} + 288 T^{3} - 7633 T^{4} + 288 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^3$ \( 1 - 83 T^{2} + 648 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^3$ \( 1 - 8878 T^{4} + p^{4} T^{8} \)
89$C_2^3$ \( 1 + 177 T^{2} + 23408 T^{4} + 177 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^3$ \( 1 + 8 T + 32 T^{2} - 1296 T^{3} - 14593 T^{4} - 1296 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.900994954921216854713538298264, −7.66369343652411687010580786648, −7.56061302029567500377161245750, −7.54620206228658867090595195722, −7.02253322885734794416303091653, −6.69590141912643334997691692505, −6.69233895726361220595455291453, −6.45790479362452439118583012005, −5.87435211155103522092203689032, −5.82358760543928248024885164765, −5.65661039334509123620342753139, −5.32029816331612412471456409615, −5.24270114453039340868982476978, −4.99082182462089597280116828847, −4.61227935865390320227937718770, −4.46930757873736225598182002418, −3.95743138411987513109603953640, −3.90633415054521044022320438037, −3.69699172784419110073451774031, −2.93642470467031870699449352504, −2.82000551729012040702297177174, −2.23564413369130839083879868085, −1.15353064925330880371851115560, −1.09468076754599937284568282782, −0.73731258284675205070941863614, 0.73731258284675205070941863614, 1.09468076754599937284568282782, 1.15353064925330880371851115560, 2.23564413369130839083879868085, 2.82000551729012040702297177174, 2.93642470467031870699449352504, 3.69699172784419110073451774031, 3.90633415054521044022320438037, 3.95743138411987513109603953640, 4.46930757873736225598182002418, 4.61227935865390320227937718770, 4.99082182462089597280116828847, 5.24270114453039340868982476978, 5.32029816331612412471456409615, 5.65661039334509123620342753139, 5.82358760543928248024885164765, 5.87435211155103522092203689032, 6.45790479362452439118583012005, 6.69233895726361220595455291453, 6.69590141912643334997691692505, 7.02253322885734794416303091653, 7.54620206228658867090595195722, 7.56061302029567500377161245750, 7.66369343652411687010580786648, 7.900994954921216854713538298264

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