Properties

Label 8-378e4-1.1-c1e4-0-5
Degree $8$
Conductor $20415837456$
Sign $1$
Analytic cond. $82.9995$
Root an. cond. $1.73733$
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  + 4-s + 2·7-s + 10·25-s + 2·28-s − 6·31-s + 16·37-s + 16·43-s − 11·49-s − 48·61-s − 64-s − 4·67-s + 18·73-s + 26·79-s + 10·100-s − 60·103-s + 16·109-s − 13·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.755·7-s + 2·25-s + 0.377·28-s − 1.07·31-s + 2.63·37-s + 2.43·43-s − 1.57·49-s − 6.14·61-s − 1/8·64-s − 0.488·67-s + 2.10·73-s + 2.92·79-s + 100-s − 5.91·103-s + 1.53·109-s − 1.18·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{12} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(82.9995\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.862456791\)
\(L(\frac12)\) \(\approx\) \(2.862456791\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3 \( 1 \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 + 14 T^{2} - 2013 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \)
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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

−8.086574231579406264468996357959, −7.909285770097460285166737956836, −7.77841752412998338198691453115, −7.69440705983956753760806381873, −7.26157105986856312318605980726, −7.01071852508844524120473229476, −6.55560880760996560227572417322, −6.53875476603521524901567443317, −6.41730007512408445821643461432, −5.79194262377359026604983492510, −5.74087962421612464081947245288, −5.52192614860960498756767291182, −5.10207548236249855531975780740, −4.67343337505694314375490376297, −4.59758807589712281583841461350, −4.27046357432050017177124117155, −4.17075565184571896968743572361, −3.51379620733472021601159874160, −3.04912566107652497715812973662, −3.00258585574345978194414925958, −2.70612079438598448175909728142, −2.05571940661367282407657899900, −1.82160538634149474370599481218, −1.28177715324940123000578475017, −0.71997311433045778388463728406, 0.71997311433045778388463728406, 1.28177715324940123000578475017, 1.82160538634149474370599481218, 2.05571940661367282407657899900, 2.70612079438598448175909728142, 3.00258585574345978194414925958, 3.04912566107652497715812973662, 3.51379620733472021601159874160, 4.17075565184571896968743572361, 4.27046357432050017177124117155, 4.59758807589712281583841461350, 4.67343337505694314375490376297, 5.10207548236249855531975780740, 5.52192614860960498756767291182, 5.74087962421612464081947245288, 5.79194262377359026604983492510, 6.41730007512408445821643461432, 6.53875476603521524901567443317, 6.55560880760996560227572417322, 7.01071852508844524120473229476, 7.26157105986856312318605980726, 7.69440705983956753760806381873, 7.77841752412998338198691453115, 7.909285770097460285166737956836, 8.086574231579406264468996357959

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