Properties

Label 8-252e4-1.1-c1e4-0-7
Degree $8$
Conductor $4032758016$
Sign $1$
Analytic cond. $16.3949$
Root an. cond. $1.41853$
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  + 2·4-s + 2·7-s + 10·19-s − 2·25-s + 4·28-s − 2·31-s − 10·37-s − 11·49-s + 24·61-s − 8·64-s + 30·67-s + 6·73-s + 20·76-s − 6·79-s − 4·100-s − 2·103-s − 10·109-s + 10·121-s − 4·124-s + 127-s + 131-s + 20·133-s + 137-s + 139-s − 20·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 4-s + 0.755·7-s + 2.29·19-s − 2/5·25-s + 0.755·28-s − 0.359·31-s − 1.64·37-s − 1.57·49-s + 3.07·61-s − 64-s + 3.66·67-s + 0.702·73-s + 2.29·76-s − 0.675·79-s − 2/5·100-s − 0.197·103-s − 0.957·109-s + 0.909·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 1.73·133-s + 0.0854·137-s + 0.0848·139-s − 1.64·148-s + 0.0819·149-s + 0.0813·151-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: $1$
Analytic conductor: \(16.3949\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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

\(L(1)\) \(\approx\) \(2.640930178\)
\(L(\frac12)\) \(\approx\) \(2.640930178\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
good5$C_2^3$ \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^3$ \( 1 + 26 T^{2} + 387 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 38 T^{2} + 915 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 59 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 70 T^{2} + 2691 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 82 T^{2} + 3915 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 50 T^{2} - 5421 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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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

−8.807321997834808029565343629984, −8.280999493335362395725858734036, −8.193456821518421881085677547807, −8.014557099205812013206611133096, −7.964669319577384183043744636852, −7.26283370374467443444372508094, −7.17231928775559236057656197845, −6.99773391736015001621557979536, −6.76898018964996469565361873738, −6.60361457132136596147772661406, −5.94944676795350902916491693686, −5.81462826645787011471499938356, −5.59583210540170755716544950356, −5.03638635347303934260229228262, −4.98205163891639825919455148998, −4.94391502063436488693379766407, −4.17662701268955995180043006534, −3.80241602635471355342087179505, −3.62182210638868976049755199259, −3.18723017776941444008426918140, −2.88612934852103026029688503894, −2.23923791318654368970529960781, −2.07424347127167627132014105087, −1.53802630461551103844380001748, −0.902949173985685233608125195250, 0.902949173985685233608125195250, 1.53802630461551103844380001748, 2.07424347127167627132014105087, 2.23923791318654368970529960781, 2.88612934852103026029688503894, 3.18723017776941444008426918140, 3.62182210638868976049755199259, 3.80241602635471355342087179505, 4.17662701268955995180043006534, 4.94391502063436488693379766407, 4.98205163891639825919455148998, 5.03638635347303934260229228262, 5.59583210540170755716544950356, 5.81462826645787011471499938356, 5.94944676795350902916491693686, 6.60361457132136596147772661406, 6.76898018964996469565361873738, 6.99773391736015001621557979536, 7.17231928775559236057656197845, 7.26283370374467443444372508094, 7.964669319577384183043744636852, 8.014557099205812013206611133096, 8.193456821518421881085677547807, 8.280999493335362395725858734036, 8.807321997834808029565343629984

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