Properties

Label 8-560e4-1.1-c7e4-0-0
Degree $8$
Conductor $98344960000$
Sign $1$
Analytic cond. $9.36511\times 10^{8}$
Root an. cond. $13.2263$
Motivic weight $7$
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  + 37·3-s − 500·5-s + 1.37e3·7-s − 2.49e3·9-s + 7.22e3·11-s + 7.95e3·13-s − 1.85e4·15-s + 6.47e4·17-s + 3.78e4·19-s + 5.07e4·21-s − 1.11e5·23-s + 1.56e5·25-s − 7.20e4·27-s + 2.18e5·29-s − 2.19e5·31-s + 2.67e5·33-s − 6.86e5·35-s − 2.12e5·37-s + 2.94e5·39-s − 1.21e6·41-s − 1.07e6·43-s + 1.24e6·45-s − 1.10e5·47-s + 1.17e6·49-s + 2.39e6·51-s + 3.56e6·53-s − 3.61e6·55-s + ⋯
L(s)  = 1  + 0.791·3-s − 1.78·5-s + 1.51·7-s − 1.14·9-s + 1.63·11-s + 1.00·13-s − 1.41·15-s + 3.19·17-s + 1.26·19-s + 1.19·21-s − 1.91·23-s + 2·25-s − 0.704·27-s + 1.66·29-s − 1.32·31-s + 1.29·33-s − 2.70·35-s − 0.690·37-s + 0.794·39-s − 2.75·41-s − 2.06·43-s + 2.03·45-s − 0.155·47-s + 10/7·49-s + 2.52·51-s + 3.29·53-s − 2.92·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(9.36511\times 10^{8}\)
Root analytic conductor: \(13.2263\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(12.94626533\)
\(L(\frac12)\) \(\approx\) \(12.94626533\)
\(L(\frac{9}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + p^{3} T )^{4} \)
7$C_1$ \( ( 1 - p^{3} T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 37 T + 3863 T^{2} - 54388 p T^{3} + 808672 p^{2} T^{4} - 54388 p^{8} T^{5} + 3863 p^{14} T^{6} - 37 p^{21} T^{7} + p^{28} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 7225 T + 34600123 T^{2} - 49257053492 T^{3} + 125572429271028 T^{4} - 49257053492 p^{7} T^{5} + 34600123 p^{14} T^{6} - 7225 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 7957 T + 78765741 T^{2} - 692449383218 T^{3} + 9436293809535266 T^{4} - 692449383218 p^{7} T^{5} + 78765741 p^{14} T^{6} - 7957 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 64727 T + 2141773081 T^{2} - 47741760670114 T^{3} + 955964193710065206 T^{4} - 47741760670114 p^{7} T^{5} + 2141773081 p^{14} T^{6} - 64727 p^{21} T^{7} + p^{28} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 37866 T + 2093244216 T^{2} - 64496367872706 T^{3} + 2883596566634786846 T^{4} - 64496367872706 p^{7} T^{5} + 2093244216 p^{14} T^{6} - 37866 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 111470 T + 12352957192 T^{2} + 660048723178798 T^{3} + 47837838854772659214 T^{4} + 660048723178798 p^{7} T^{5} + 12352957192 p^{14} T^{6} + 111470 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 218711 T + 56606470061 T^{2} - 6047870884766846 T^{3} + \)\(10\!\cdots\!46\)\( T^{4} - 6047870884766846 p^{7} T^{5} + 56606470061 p^{14} T^{6} - 218711 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 219348 T + 74650121836 T^{2} + 12768400485861444 T^{3} + \)\(31\!\cdots\!50\)\( T^{4} + 12768400485861444 p^{7} T^{5} + 74650121836 p^{14} T^{6} + 219348 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 212844 T + 357803238100 T^{2} + 60507418655778372 T^{3} + \)\(49\!\cdots\!18\)\( T^{4} + 60507418655778372 p^{7} T^{5} + 357803238100 p^{14} T^{6} + 212844 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 1215338 T + 1221426625564 T^{2} + 739677778209578542 T^{3} + \)\(39\!\cdots\!10\)\( T^{4} + 739677778209578542 p^{7} T^{5} + 1221426625564 p^{14} T^{6} + 1215338 p^{21} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 1074782 T + 1241352511200 T^{2} + 789348415564299766 T^{3} + \)\(52\!\cdots\!02\)\( T^{4} + 789348415564299766 p^{7} T^{5} + 1241352511200 p^{14} T^{6} + 1074782 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 110699 T + 262265536967 T^{2} - 318211163910969976 T^{3} + \)\(40\!\cdots\!40\)\( T^{4} - 318211163910969976 p^{7} T^{5} + 262265536967 p^{14} T^{6} + 110699 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 3568154 T + 6826452592684 T^{2} - 9289868635999433974 T^{3} + \)\(10\!\cdots\!90\)\( T^{4} - 9289868635999433974 p^{7} T^{5} + 6826452592684 p^{14} T^{6} - 3568154 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 4432 p T + 4572743385932 T^{2} + 3183449149265459696 T^{3} + \)\(14\!\cdots\!94\)\( T^{4} + 3183449149265459696 p^{7} T^{5} + 4572743385932 p^{14} T^{6} + 4432 p^{22} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 5355426 T + 21559649494276 T^{2} - 55895134416232311774 T^{3} + \)\(11\!\cdots\!06\)\( T^{4} - 55895134416232311774 p^{7} T^{5} + 21559649494276 p^{14} T^{6} - 5355426 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 10352960 T + 63169987396044 T^{2} + \)\(25\!\cdots\!16\)\( T^{3} + \)\(72\!\cdots\!94\)\( T^{4} + \)\(25\!\cdots\!16\)\( p^{7} T^{5} + 63169987396044 p^{14} T^{6} + 10352960 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 3607808 T + 13350575639900 T^{2} + 33863456913253298432 T^{3} + \)\(15\!\cdots\!18\)\( T^{4} + 33863456913253298432 p^{7} T^{5} + 13350575639900 p^{14} T^{6} + 3607808 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 2787392 T + 42028451310924 T^{2} - 86824620532098338176 T^{3} + \)\(68\!\cdots\!02\)\( T^{4} - 86824620532098338176 p^{7} T^{5} + 42028451310924 p^{14} T^{6} - 2787392 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 735975 T + 52963476154071 T^{2} + 13424347275902279856 T^{3} + \)\(12\!\cdots\!56\)\( T^{4} + 13424347275902279856 p^{7} T^{5} + 52963476154071 p^{14} T^{6} - 735975 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 2001724 T + 101529783773852 T^{2} + \)\(14\!\cdots\!28\)\( T^{3} + \)\(40\!\cdots\!98\)\( T^{4} + \)\(14\!\cdots\!28\)\( p^{7} T^{5} + 101529783773852 p^{14} T^{6} + 2001724 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 13872714 T + 205852613098156 T^{2} + \)\(15\!\cdots\!94\)\( T^{3} + \)\(13\!\cdots\!86\)\( T^{4} + \)\(15\!\cdots\!94\)\( p^{7} T^{5} + 205852613098156 p^{14} T^{6} + 13872714 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 12763773 T + 247707500464233 T^{2} + \)\(17\!\cdots\!54\)\( T^{3} + \)\(23\!\cdots\!18\)\( T^{4} + \)\(17\!\cdots\!54\)\( p^{7} T^{5} + 247707500464233 p^{14} T^{6} + 12763773 p^{21} T^{7} + p^{28} T^{8} \)
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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

−6.86539238813055299662752984746, −6.27812175628740981606478059068, −6.04672335118437371776690183562, −5.79159190983087613195388468699, −5.70739043131371161358308237020, −5.26828571567099193850983462860, −5.12562298993272082108695696090, −5.07136006745792923759484768237, −4.55476934899950493094188144540, −4.23883805679458414542479468944, −3.94394246419865217664164790168, −3.78186467261725826577101879809, −3.77225972394118537611783592393, −3.23172203743407960902110160045, −3.11380551678898809583605452670, −3.01602179083818647016498146707, −2.82615963509794397852333038862, −2.04995256996039130469514101759, −1.82697321779635346699774028784, −1.57071425172836848094745196692, −1.45289810492537162330664638238, −1.06852215483495038520973049209, −0.77698664733176010856605735660, −0.48798720490749348711065679227, −0.35285342234381823969648840141, 0.35285342234381823969648840141, 0.48798720490749348711065679227, 0.77698664733176010856605735660, 1.06852215483495038520973049209, 1.45289810492537162330664638238, 1.57071425172836848094745196692, 1.82697321779635346699774028784, 2.04995256996039130469514101759, 2.82615963509794397852333038862, 3.01602179083818647016498146707, 3.11380551678898809583605452670, 3.23172203743407960902110160045, 3.77225972394118537611783592393, 3.78186467261725826577101879809, 3.94394246419865217664164790168, 4.23883805679458414542479468944, 4.55476934899950493094188144540, 5.07136006745792923759484768237, 5.12562298993272082108695696090, 5.26828571567099193850983462860, 5.70739043131371161358308237020, 5.79159190983087613195388468699, 6.04672335118437371776690183562, 6.27812175628740981606478059068, 6.86539238813055299662752984746

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