Properties

Label 8-3e16-1.1-c8e4-0-1
Degree $8$
Conductor $43046721$
Sign $1$
Analytic cond. $1.18558\times 10^{6}$
Root an. cond. $5.74435$
Motivic weight $8$
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  + 352·4-s − 3.93e3·7-s + 9.10e4·13-s + 6.55e4·16-s + 6.09e5·19-s − 1.03e5·25-s − 1.38e6·28-s + 3.28e5·31-s − 2.65e6·37-s − 1.15e6·43-s + 1.53e7·49-s + 3.20e7·52-s + 3.84e7·61-s + 2.55e7·64-s + 1.19e6·67-s + 5.14e7·73-s + 2.14e8·76-s + 4.71e7·79-s − 3.58e8·91-s − 2.72e8·97-s − 3.65e7·100-s − 6.71e7·103-s − 4.04e8·109-s − 2.57e8·112-s − 2.70e8·121-s + 1.15e8·124-s + 127-s + ⋯
L(s)  = 1  + 11/8·4-s − 1.63·7-s + 3.18·13-s + 16-s + 4.67·19-s − 0.265·25-s − 2.25·28-s + 0.355·31-s − 1.41·37-s − 0.336·43-s + 2.67·49-s + 4.38·52-s + 2.77·61-s + 1.52·64-s + 0.0593·67-s + 1.80·73-s + 6.43·76-s + 1.21·79-s − 5.22·91-s − 3.08·97-s − 0.365·100-s − 0.596·103-s − 2.86·109-s − 1.63·112-s − 1.26·121-s + 0.489·124-s − 7.66·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(43046721\)    =    \(3^{16}\)
Sign: $1$
Analytic conductor: \(1.18558\times 10^{6}\)
Root analytic conductor: \(5.74435\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 43046721,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(9.761921593\)
\(L(\frac12)\) \(\approx\) \(9.761921593\)
\(L(5)\) 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 \( 1 \)
good2$C_2^3$ \( 1 - 11 p^{5} T^{2} + 57 p^{10} T^{4} - 11 p^{21} T^{6} + p^{32} T^{8} \)
5$C_2^3$ \( 1 + 103874 T^{2} - 141798082749 T^{4} + 103874 p^{16} T^{6} + p^{32} T^{8} \)
7$C_2^2$ \( ( 1 + 281 p T - 38688 p^{2} T^{2} + 281 p^{9} T^{3} + p^{16} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 270446786 T^{2} + 27191734194157635 T^{4} + 270446786 p^{16} T^{6} + p^{32} T^{8} \)
13$C_2^2$ \( ( 1 - 45505 T + 1254974304 T^{2} - 45505 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 10398069506 T^{2} + p^{16} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8021 p T + p^{8} T^{2} )^{4} \)
23$C_2^3$ \( 1 + 139373917826 T^{2} + \)\(13\!\cdots\!15\)\( T^{4} + 139373917826 p^{16} T^{6} + p^{32} T^{8} \)
29$C_2^3$ \( 1 + 654062888258 T^{2} + \)\(17\!\cdots\!43\)\( T^{4} + 654062888258 p^{16} T^{6} + p^{32} T^{8} \)
31$C_2^2$ \( ( 1 - 164350 T - 825880114941 T^{2} - 164350 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 663937 T + p^{8} T^{2} )^{4} \)
41$C_2^3$ \( 1 + 15089973815426 T^{2} + \)\(16\!\cdots\!35\)\( T^{4} + 15089973815426 p^{16} T^{6} + p^{32} T^{8} \)
43$C_2^2$ \( ( 1 + 575330 T - 11357195668701 T^{2} + 575330 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 37649036054782 T^{2} + \)\(85\!\cdots\!03\)\( T^{4} - 37649036054782 p^{16} T^{6} + p^{32} T^{8} \)
53$C_2^2$ \( ( 1 - 16847444857538 T^{2} + p^{16} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 268260543906242 T^{2} + \)\(50\!\cdots\!23\)\( T^{4} + 268260543906242 p^{16} T^{6} + p^{32} T^{8} \)
61$C_2^2$ \( ( 1 - 19212961 T + 177430557390240 T^{2} - 19212961 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 598033 T - 405710034087552 T^{2} - 598033 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 434653057247618 T^{2} + p^{16} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 12850175 T + p^{8} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 23584657 T - 960872764098912 T^{2} - 23584657 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 3385619252644418 T^{2} + \)\(63\!\cdots\!43\)\( T^{4} + 3385619252644418 p^{16} T^{6} + p^{32} T^{8} \)
89$C_2^2$ \( ( 1 - 892961548706 p^{2} T^{2} + p^{16} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 136489631 T + 10791985776139200 T^{2} + 136489631 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
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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

−8.877834457267732237915167364051, −8.598319786292773944480925770115, −8.298272524065041022044410712408, −8.047282933557112648564933310136, −7.59826739943002681322504095727, −7.24386836750511443779637780404, −6.96346970266659046997590332683, −6.69798683693121050367284845278, −6.66325230025739260424999353779, −5.95775917723816383732082545175, −5.79737468239884109229724063311, −5.68816233429334326142059798623, −5.13146208090468048727092180639, −5.02203283637672483115106277881, −3.88115359811770409589871973665, −3.73500703412212177874436139095, −3.47718258096914453659246642738, −3.43524444481153067024501430182, −2.76257429967135427722207774630, −2.61336520083607999854862484340, −1.99399241660350366866713959910, −1.24256078698288766411791136047, −1.11637042076213741673761342588, −1.03079640261614110387482771198, −0.38883560382088629746744857623, 0.38883560382088629746744857623, 1.03079640261614110387482771198, 1.11637042076213741673761342588, 1.24256078698288766411791136047, 1.99399241660350366866713959910, 2.61336520083607999854862484340, 2.76257429967135427722207774630, 3.43524444481153067024501430182, 3.47718258096914453659246642738, 3.73500703412212177874436139095, 3.88115359811770409589871973665, 5.02203283637672483115106277881, 5.13146208090468048727092180639, 5.68816233429334326142059798623, 5.79737468239884109229724063311, 5.95775917723816383732082545175, 6.66325230025739260424999353779, 6.69798683693121050367284845278, 6.96346970266659046997590332683, 7.24386836750511443779637780404, 7.59826739943002681322504095727, 8.047282933557112648564933310136, 8.298272524065041022044410712408, 8.598319786292773944480925770115, 8.877834457267732237915167364051

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