Properties

Label 8-21e8-1.1-c1e4-0-2
Degree $8$
Conductor $37822859361$
Sign $1$
Analytic cond. $153.766$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 4·16-s − 6·19-s + 4·25-s − 6·31-s + 2·37-s − 4·43-s + 12·61-s − 16·64-s − 22·67-s − 6·73-s + 12·76-s − 10·79-s − 8·100-s + 30·103-s + 2·109-s − 20·121-s + 12·124-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 4-s + 16-s − 1.37·19-s + 4/5·25-s − 1.07·31-s + 0.328·37-s − 0.609·43-s + 1.53·61-s − 2·64-s − 2.68·67-s − 0.702·73-s + 1.37·76-s − 1.12·79-s − 4/5·100-s + 2.95·103-s + 0.191·109-s − 1.81·121-s + 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.328·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(153.766\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8149293798\)
\(L(\frac12)\) \(\approx\) \(0.8149293798\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2$$\times$$C_2^2$ \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \)
5$C_2^3$ \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 + 20 T^{2} + 279 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^3$ \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 + 56 T^{2} + 927 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 98 T^{2} + 6795 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
71$C_2^2$ \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 154 T^{2} + 15795 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
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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

−8.168184551169936499474265486714, −8.036525641795057330386969831790, −7.31882673586797768926991150514, −7.31126099192007951128571652518, −7.19803803538844416118098401750, −7.04536993746579495464528158901, −6.41021049900033197763884062021, −6.16455411294987780444178176634, −6.13431329155578752092901838130, −5.87888561710092322925668569894, −5.41715552751071598966957061792, −5.29657453434507699830714010670, −4.95541593508220290391223849567, −4.51446094399531861215859291780, −4.48891256186800905268486595711, −4.26747934783780615839524012952, −3.88273970107117388645674314634, −3.43788511376184162974423149266, −3.19440087598142994601714977897, −3.07071901464324400837511754963, −2.32465408528538339999183602548, −2.20184535836084874211972049644, −1.58126299528232681659487632190, −1.19273837477788576291842007937, −0.35547448536739301371692184644, 0.35547448536739301371692184644, 1.19273837477788576291842007937, 1.58126299528232681659487632190, 2.20184535836084874211972049644, 2.32465408528538339999183602548, 3.07071901464324400837511754963, 3.19440087598142994601714977897, 3.43788511376184162974423149266, 3.88273970107117388645674314634, 4.26747934783780615839524012952, 4.48891256186800905268486595711, 4.51446094399531861215859291780, 4.95541593508220290391223849567, 5.29657453434507699830714010670, 5.41715552751071598966957061792, 5.87888561710092322925668569894, 6.13431329155578752092901838130, 6.16455411294987780444178176634, 6.41021049900033197763884062021, 7.04536993746579495464528158901, 7.19803803538844416118098401750, 7.31126099192007951128571652518, 7.31882673586797768926991150514, 8.036525641795057330386969831790, 8.168184551169936499474265486714

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