Properties

Label 8-384e4-1.1-c3e4-0-8
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 18·9-s + 576·23-s − 436·25-s + 2.30e3·47-s + 796·49-s + 2.88e3·71-s − 712·73-s − 405·81-s + 2.60e3·97-s + 3.52e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.58e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2/3·9-s + 5.22·23-s − 3.48·25-s + 7.15·47-s + 2.32·49-s + 4.81·71-s − 1.14·73-s − 5/9·81-s + 2.72·97-s + 2.64·121-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 − 0.719·169-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

\(L(2)\) \(\approx\) \(9.247685320\)
\(L(\frac12)\) \(\approx\) \(9.247685320\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 2 p^{2} T^{2} + p^{6} T^{4} \)
good5$C_2^2$ \( ( 1 + 218 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 398 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 1762 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 790 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 7234 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 13070 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2$ \( ( 1 - 144 T + p^{3} T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 48746 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 10910 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 96122 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 44530 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 106526 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2$ \( ( 1 - 576 T + p^{3} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 32762 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 239362 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 199946 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 21202 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 720 T + p^{3} T^{2} )^{4} \)
73$C_2$ \( ( 1 + 178 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 50366 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 951730 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 660850 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 650 T + p^{3} T^{2} )^{4} \)
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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

−7.52249204921183301390981186838, −7.33650465607363445279330073571, −7.22435859094801183596518224116, −7.16950523962271956946904431224, −7.14581181596770401290547447106, −6.46934104582836833493853256325, −6.21080290157595013091644441308, −5.86257569031097514380175300652, −5.81841321277477143207322227244, −5.43196632230117726032954602730, −5.19581953562491948996588353346, −4.84787677209530728715124635955, −4.78427528936009213613730344083, −4.08150836603429411347850573859, −4.01715966982008083902243858974, −3.81919092208375403667953494505, −3.57526767602722731241816635969, −2.92908954917016416418303471799, −2.69523285749397244256011972089, −2.39245457328914294874503367372, −2.11873929994126514290721565678, −1.61474958555642258430873526582, −0.873310423537784602515782137560, −0.76134128497795802427477836283, −0.69381515492074509838449375135, 0.69381515492074509838449375135, 0.76134128497795802427477836283, 0.873310423537784602515782137560, 1.61474958555642258430873526582, 2.11873929994126514290721565678, 2.39245457328914294874503367372, 2.69523285749397244256011972089, 2.92908954917016416418303471799, 3.57526767602722731241816635969, 3.81919092208375403667953494505, 4.01715966982008083902243858974, 4.08150836603429411347850573859, 4.78427528936009213613730344083, 4.84787677209530728715124635955, 5.19581953562491948996588353346, 5.43196632230117726032954602730, 5.81841321277477143207322227244, 5.86257569031097514380175300652, 6.21080290157595013091644441308, 6.46934104582836833493853256325, 7.14581181596770401290547447106, 7.16950523962271956946904431224, 7.22435859094801183596518224116, 7.33650465607363445279330073571, 7.52249204921183301390981186838

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