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Properties

Label 8-20e4-1.1-c4e4-0-0
Degree $8$
Conductor $160000$
Sign $1$
Analytic cond. $18.2682$
Root an. cond. $1.43784$
Motivic weight $4$
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  − 10·3-s − 6·5-s + 110·7-s + 50·9-s − 300·11-s − 360·13-s + 60·15-s + 960·17-s − 1.10e3·21-s − 810·23-s + 946·25-s + 270·27-s − 836·31-s + 3.00e3·33-s − 660·35-s − 660·37-s + 3.60e3·39-s + 2.96e3·41-s + 3.27e3·43-s − 300·45-s − 2.25e3·47-s + 6.05e3·49-s − 9.60e3·51-s + 1.98e3·53-s + 1.80e3·55-s + 1.58e4·61-s + 5.50e3·63-s + ⋯
L(s)  = 1  − 1.11·3-s − 0.239·5-s + 2.24·7-s + 0.617·9-s − 2.47·11-s − 2.13·13-s + 4/15·15-s + 3.32·17-s − 2.49·21-s − 1.53·23-s + 1.51·25-s + 0.370·27-s − 0.869·31-s + 2.75·33-s − 0.538·35-s − 0.482·37-s + 2.36·39-s + 1.76·41-s + 1.76·43-s − 0.148·45-s − 1.01·47-s + 2.51·49-s − 3.69·51-s + 0.704·53-s + 0.595·55-s + 4.25·61-s + 1.38·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(18.2682\)
Root analytic conductor: \(1.43784\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 160000,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.177121367\)
\(L(\frac12)\) \(\approx\) \(1.177121367\)
\(L(3)\) 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$C_2^2$ \( 1 + 6 T - 182 p T^{2} + 6 p^{4} T^{3} + p^{8} T^{4} \)
good3$D_4\times C_2$ \( 1 + 10 T + 50 T^{2} - 10 p^{3} T^{3} - 14 p^{6} T^{4} - 10 p^{7} T^{5} + 50 p^{8} T^{6} + 10 p^{12} T^{7} + p^{16} T^{8} \)
7$D_4\times C_2$ \( 1 - 110 T + 6050 T^{2} - 311190 T^{3} + 15823298 T^{4} - 311190 p^{4} T^{5} + 6050 p^{8} T^{6} - 110 p^{12} T^{7} + p^{16} T^{8} \)
11$D_{4}$ \( ( 1 + 150 T + 28882 T^{2} + 150 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 360 T + 64800 T^{2} + 14552280 T^{3} + 3127330814 T^{4} + 14552280 p^{4} T^{5} + 64800 p^{8} T^{6} + 360 p^{12} T^{7} + p^{16} T^{8} \)
17$D_4\times C_2$ \( 1 - 960 T + 460800 T^{2} - 168098880 T^{3} + 52934858494 T^{4} - 168098880 p^{4} T^{5} + 460800 p^{8} T^{6} - 960 p^{12} T^{7} + p^{16} T^{8} \)
19$D_4\times C_2$ \( 1 - 79796 T^{2} + 32223922086 T^{4} - 79796 p^{8} T^{6} + p^{16} T^{8} \)
23$D_4\times C_2$ \( 1 + 810 T + 328050 T^{2} + 264893490 T^{3} + 211669226338 T^{4} + 264893490 p^{4} T^{5} + 328050 p^{8} T^{6} + 810 p^{12} T^{7} + p^{16} T^{8} \)
29$D_4\times C_2$ \( 1 - 1448516 T^{2} + 1055067026886 T^{4} - 1448516 p^{8} T^{6} + p^{16} T^{8} \)
31$D_{4}$ \( ( 1 + 418 T + 535098 T^{2} + 418 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 660 T + 217800 T^{2} + 345245340 T^{3} - 1278103400242 T^{4} + 345245340 p^{4} T^{5} + 217800 p^{8} T^{6} + 660 p^{12} T^{7} + p^{16} T^{8} \)
41$D_{4}$ \( ( 1 - 1482 T + 6194578 T^{2} - 1482 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 3270 T + 5346450 T^{2} - 11205966270 T^{3} + 23487235317602 T^{4} - 11205966270 p^{4} T^{5} + 5346450 p^{8} T^{6} - 3270 p^{12} T^{7} + p^{16} T^{8} \)
47$D_4\times C_2$ \( 1 + 2250 T + 2531250 T^{2} + 7891319250 T^{3} + 22718088596834 T^{4} + 7891319250 p^{4} T^{5} + 2531250 p^{8} T^{6} + 2250 p^{12} T^{7} + p^{16} T^{8} \)
53$D_4\times C_2$ \( 1 - 1980 T + 1960200 T^{2} - 16406396820 T^{3} + 137161065548878 T^{4} - 16406396820 p^{4} T^{5} + 1960200 p^{8} T^{6} - 1980 p^{12} T^{7} + p^{16} T^{8} \)
59$D_4\times C_2$ \( 1 + 162508 T^{2} + 44050763806758 T^{4} + 162508 p^{8} T^{6} + p^{16} T^{8} \)
61$D_{4}$ \( ( 1 - 7914 T + 36788306 T^{2} - 7914 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 4810 T + 11568050 T^{2} + 108537019890 T^{3} + 1012520446566818 T^{4} + 108537019890 p^{4} T^{5} + 11568050 p^{8} T^{6} + 4810 p^{12} T^{7} + p^{16} T^{8} \)
71$D_{4}$ \( ( 1 + 2994 T + 49877146 T^{2} + 2994 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 14060 T + 98841800 T^{2} + 628627141380 T^{3} + 3731941946353934 T^{4} + 628627141380 p^{4} T^{5} + 98841800 p^{8} T^{6} + 14060 p^{12} T^{7} + p^{16} T^{8} \)
79$D_4\times C_2$ \( 1 - 127811332 T^{2} + 7111883058262278 T^{4} - 127811332 p^{8} T^{6} + p^{16} T^{8} \)
83$D_4\times C_2$ \( 1 - 19950 T + 199001250 T^{2} - 1392338210550 T^{3} + 9242910030806018 T^{4} - 1392338210550 p^{4} T^{5} + 199001250 p^{8} T^{6} - 19950 p^{12} T^{7} + p^{16} T^{8} \)
89$D_4\times C_2$ \( 1 - 213169796 T^{2} + 19233126298796166 T^{4} - 213169796 p^{8} T^{6} + p^{16} T^{8} \)
97$D_4\times C_2$ \( 1 + 3180 T + 5056200 T^{2} - 122563753980 T^{3} - 13176145025425522 T^{4} - 122563753980 p^{4} T^{5} + 5056200 p^{8} T^{6} + 3180 p^{12} T^{7} + p^{16} T^{8} \)
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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

−12.85525477438734264211129749235, −12.55921310098960846433001094683, −12.50771465313888285192842780525, −11.98177362468069521849504088685, −11.69413535172765885515997927402, −11.51795364341511175963984329506, −10.84626697741693508379457614738, −10.47671239983168355273154478960, −10.41626048719966863896871139575, −10.05573377737995212466107089243, −9.603457126009498398754569936812, −8.916423700623931333647762270455, −8.180946595242949863408741200490, −7.955257995899469966867070290165, −7.69255358505708850579176025135, −7.44543570724354338097910072290, −6.96602763299660676122588486471, −5.72602083199864542985854354573, −5.48922813216259480015290794832, −5.31063784343133071597762670867, −4.83000559344715022310561422627, −4.27760945264762824323164603140, −2.97650759875887709875893004600, −2.18162554790584242368735364044, −0.76949609447508041604962811046, 0.76949609447508041604962811046, 2.18162554790584242368735364044, 2.97650759875887709875893004600, 4.27760945264762824323164603140, 4.83000559344715022310561422627, 5.31063784343133071597762670867, 5.48922813216259480015290794832, 5.72602083199864542985854354573, 6.96602763299660676122588486471, 7.44543570724354338097910072290, 7.69255358505708850579176025135, 7.955257995899469966867070290165, 8.180946595242949863408741200490, 8.916423700623931333647762270455, 9.603457126009498398754569936812, 10.05573377737995212466107089243, 10.41626048719966863896871139575, 10.47671239983168355273154478960, 10.84626697741693508379457614738, 11.51795364341511175963984329506, 11.69413535172765885515997927402, 11.98177362468069521849504088685, 12.50771465313888285192842780525, 12.55921310098960846433001094683, 12.85525477438734264211129749235

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