Properties

Label 8-855e4-1.1-c1e4-0-1
Degree $8$
Conductor $534397550625$
Sign $1$
Analytic cond. $2172.56$
Root an. cond. $2.61289$
Motivic weight $1$
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  + 2·2-s + 3·4-s − 2·5-s − 4·7-s + 2·8-s − 4·10-s − 2·13-s − 8·14-s − 8·17-s + 16·19-s − 6·20-s + 4·23-s + 25-s − 4·26-s − 12·28-s + 8·29-s − 20·31-s − 6·32-s − 16·34-s + 8·35-s − 20·37-s + 32·38-s − 4·40-s − 10·43-s + 8·46-s − 12·47-s − 2·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.51·7-s + 0.707·8-s − 1.26·10-s − 0.554·13-s − 2.13·14-s − 1.94·17-s + 3.67·19-s − 1.34·20-s + 0.834·23-s + 1/5·25-s − 0.784·26-s − 2.26·28-s + 1.48·29-s − 3.59·31-s − 1.06·32-s − 2.74·34-s + 1.35·35-s − 3.28·37-s + 5.19·38-s − 0.632·40-s − 1.52·43-s + 1.17·46-s − 1.75·47-s − 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(2172.56\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4771373317\)
\(L(\frac12)\) \(\approx\) \(0.4771373317\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 2 T - 15 T^{2} - 14 T^{3} + 140 T^{4} - 14 p T^{5} - 15 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 8 T + 22 T^{2} + 64 T^{3} + 387 T^{4} + 64 p T^{5} + 22 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_4\times C_2$ \( 1 - 4 T - 26 T^{2} + 16 T^{3} + 867 T^{4} + 16 p T^{5} - 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 8 T + 22 T^{2} + 128 T^{3} - 933 T^{4} + 128 p T^{5} + 22 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
37$D_{4}$ \( ( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^3$ \( 1 - 74 T^{2} + 3795 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 10 T - 3 T^{2} + 170 T^{3} + 4460 T^{4} + 170 p T^{5} - 3 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 12 T + 22 T^{2} + 336 T^{3} + 5907 T^{4} + 336 p T^{5} + 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 6 T + 33 T^{2} - 714 T^{3} - 5908 T^{4} - 714 p T^{5} + 33 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 6 T - 35 T^{2} + 378 T^{3} - 1860 T^{4} + 378 p T^{5} - 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 10 T + 29 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 2 T - 71 T^{2} - 142 T^{3} + 4 T^{4} - 142 p T^{5} - 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 18 T + 117 T^{2} - 882 T^{3} + 11012 T^{4} - 882 p T^{5} + 117 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
89$D_4\times C_2$ \( 1 - 8 T - 58 T^{2} + 448 T^{3} + 1267 T^{4} + 448 p T^{5} - 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.98640162379484144943588917881, −6.98427089782899625206182493341, −6.78177968712170771112649359570, −6.72688556460919425717174805658, −6.40476831805804803909153137550, −6.40437767461734359444151609195, −5.71111025177151462283641598065, −5.55215288811809189456164127551, −5.47604792205812285833211971385, −5.07077938489908334972766734758, −4.91284907153272873165746223580, −4.89282408308823625802978593287, −4.66718936476540772382927057986, −3.92471164930328643949208812155, −3.83131101300964818251869368282, −3.47933169873033164158730829236, −3.43197614292778025536600580339, −3.25065695249925714829790988553, −3.05356233117805522530622085727, −2.77931632868215756296614681487, −2.05070944851746593685893097010, −1.99465804068466328322066550493, −1.65939686926094256625508042628, −0.918218696833690515175329225242, −0.14091225595846314830869586172, 0.14091225595846314830869586172, 0.918218696833690515175329225242, 1.65939686926094256625508042628, 1.99465804068466328322066550493, 2.05070944851746593685893097010, 2.77931632868215756296614681487, 3.05356233117805522530622085727, 3.25065695249925714829790988553, 3.43197614292778025536600580339, 3.47933169873033164158730829236, 3.83131101300964818251869368282, 3.92471164930328643949208812155, 4.66718936476540772382927057986, 4.89282408308823625802978593287, 4.91284907153272873165746223580, 5.07077938489908334972766734758, 5.47604792205812285833211971385, 5.55215288811809189456164127551, 5.71111025177151462283641598065, 6.40437767461734359444151609195, 6.40476831805804803909153137550, 6.72688556460919425717174805658, 6.78177968712170771112649359570, 6.98427089782899625206182493341, 6.98640162379484144943588917881

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