Properties

Label 8-384e4-1.1-c3e4-0-0
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $263505.$
Root an. cond. $4.75990$
Motivic weight $3$
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  + 50·9-s + 352·23-s − 292·25-s − 1.53e3·47-s + 1.16e3·49-s − 2.72e3·71-s + 1.68e3·73-s + 1.77e3·81-s − 4.24e3·97-s + 1.09e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.91e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1.85·9-s + 3.19·23-s − 2.33·25-s − 4.76·47-s + 3.39·49-s − 4.54·71-s + 2.70·73-s + 2.42·81-s − 4.44·97-s + 0.820·121-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 + 2.23·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(263505.\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.01102213192\)
\(L(\frac12)\) \(\approx\) \(0.01102213192\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 - 50 T^{2} + p^{6} T^{4} \)
good5$C_2^2$ \( ( 1 + 146 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 582 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 546 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 2458 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 9410 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 13614 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2$ \( ( 1 - 88 T + p^{3} T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 16222 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 13718 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 8918 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 101774 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 103998 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2$ \( ( 1 + 384 T + p^{3} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 87154 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 13858 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 398266 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 598926 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 680 T + p^{3} T^{2} )^{4} \)
73$C_2$ \( ( 1 - 422 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 431862 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 1108978 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 490994 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 1062 T + p^{3} T^{2} )^{4} \)
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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

−7.59686438100540575256188407735, −7.43854969840887024066672045858, −7.25288895381699309353465038392, −7.06062002849431372424346435962, −6.96451130113298129288319355063, −6.44287444977830475843816596112, −6.38999803452639275995949366759, −6.04306176807895430099632048328, −5.62282530966299137416364506657, −5.51095416516268299849557402167, −5.04682181081487813852042669515, −4.86484764538757352230145158452, −4.67686447436466987560723384350, −4.32133106998167298688773328805, −4.02536752834519357545840212410, −3.76004204407560799852231956181, −3.33823670170944029504692060103, −3.25253909034725465449400694267, −2.56720712679993355016492086362, −2.51582880932366228807187319469, −1.85910062992479554928979402842, −1.56101234903629888423164585620, −1.11440777704256108979620738550, −1.06355969408300547519815520108, −0.01506260551987765355658224325, 0.01506260551987765355658224325, 1.06355969408300547519815520108, 1.11440777704256108979620738550, 1.56101234903629888423164585620, 1.85910062992479554928979402842, 2.51582880932366228807187319469, 2.56720712679993355016492086362, 3.25253909034725465449400694267, 3.33823670170944029504692060103, 3.76004204407560799852231956181, 4.02536752834519357545840212410, 4.32133106998167298688773328805, 4.67686447436466987560723384350, 4.86484764538757352230145158452, 5.04682181081487813852042669515, 5.51095416516268299849557402167, 5.62282530966299137416364506657, 6.04306176807895430099632048328, 6.38999803452639275995949366759, 6.44287444977830475843816596112, 6.96451130113298129288319355063, 7.06062002849431372424346435962, 7.25288895381699309353465038392, 7.43854969840887024066672045858, 7.59686438100540575256188407735

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