Properties

Label 8-72e4-1.1-c3e4-0-1
Degree $8$
Conductor $26873856$
Sign $1$
Analytic cond. $325.682$
Root an. cond. $2.06110$
Motivic weight $3$
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  + 6·4-s + 40·7-s − 28·16-s + 420·25-s + 240·28-s + 248·31-s − 372·49-s − 552·64-s + 120·73-s + 376·79-s + 520·97-s + 2.52e3·100-s − 2.28e3·103-s − 1.12e3·112-s + 2.44e3·121-s + 1.48e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.74e3·169-s + 173-s + ⋯
L(s)  = 1  + 3/4·4-s + 2.15·7-s − 0.437·16-s + 3.35·25-s + 1.61·28-s + 1.43·31-s − 1.08·49-s − 1.07·64-s + 0.192·73-s + 0.535·79-s + 0.544·97-s + 2.51·100-s − 2.18·103-s − 0.944·112-s + 1.83·121-s + 1.07·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.795·169-s + 0.000439·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(325.682\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 26873856,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.769942019\)
\(L(\frac12)\) \(\approx\) \(4.769942019\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 3 p T^{2} + p^{6} T^{4} \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - 42 p T^{2} + p^{6} T^{4} )^{2} \)
7$C_2$ \( ( 1 - 10 T + p^{3} T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 1222 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 874 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 4194 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 362 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 1806 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 12062 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 2 p T + p^{3} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 97786 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2958 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 144934 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 4894 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 280114 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 127482 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 168842 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 94646 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
73$C_2$ \( ( 1 - 30 T + p^{3} T^{2} )^{4} \)
79$C_2$ \( ( 1 - 94 T + p^{3} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 694134 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 846738 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 130 T + p^{3} T^{2} )^{4} \)
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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

−10.68271395759717941866855520390, −9.989467472383640914860837199538, −9.829792816275799073305332603816, −9.543421036720079125957776495723, −9.006553114226092258440059070768, −8.683527573471398514215268319860, −8.437182450692687677778447787886, −8.133353239978715158563256564496, −8.130305362130291609866836020559, −7.36016460731941364759068163279, −7.24447542527269233845541087206, −6.96759718286122523807758828622, −6.43642249899004322837223289722, −6.28495307285863210110482881597, −5.81955109009358951116203357083, −5.05073538265799350646493468185, −4.98240964575716959785315717052, −4.66818566401842876303810128580, −4.47617310568157483875422497403, −3.63486623278384392430495117248, −2.98964440745171456406232368856, −2.73163268129322810587188564436, −1.95698788669787030287002228668, −1.50441641866594343879103180379, −0.843581455303045577154825550162, 0.843581455303045577154825550162, 1.50441641866594343879103180379, 1.95698788669787030287002228668, 2.73163268129322810587188564436, 2.98964440745171456406232368856, 3.63486623278384392430495117248, 4.47617310568157483875422497403, 4.66818566401842876303810128580, 4.98240964575716959785315717052, 5.05073538265799350646493468185, 5.81955109009358951116203357083, 6.28495307285863210110482881597, 6.43642249899004322837223289722, 6.96759718286122523807758828622, 7.24447542527269233845541087206, 7.36016460731941364759068163279, 8.130305362130291609866836020559, 8.133353239978715158563256564496, 8.437182450692687677778447787886, 8.683527573471398514215268319860, 9.006553114226092258440059070768, 9.543421036720079125957776495723, 9.829792816275799073305332603816, 9.989467472383640914860837199538, 10.68271395759717941866855520390

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