Properties

Label 8-1400e4-1.1-c1e4-0-1
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $15617.8$
Root an. cond. $3.34350$
Motivic weight $1$
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  + 3·9-s − 8·11-s − 6·19-s − 22·29-s − 20·31-s − 18·41-s − 2·49-s + 4·59-s + 28·61-s − 18·71-s − 2·79-s − 7·81-s + 2·89-s − 24·99-s − 22·109-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s − 18·171-s + ⋯
L(s)  = 1  + 9-s − 2.41·11-s − 1.37·19-s − 4.08·29-s − 3.59·31-s − 2.81·41-s − 2/7·49-s + 0.520·59-s + 3.58·61-s − 2.13·71-s − 0.225·79-s − 7/9·81-s + 0.211·89-s − 2.41·99-s − 2.10·109-s + 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s − 1.37·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(15617.8\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1918077694\)
\(L(\frac12)\) \(\approx\) \(0.1918077694\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 59 T^{2} + 1444 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 59 T^{2} + 1720 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 69 T^{2} + 2972 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 123 T^{2} + 7136 T^{4} - 123 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 160 T^{2} + 11406 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 54 T^{2} - 85 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 139 T^{2} + 14260 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + T + 120 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 143 T^{2} + 10284 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.83488907603964730613411195985, −6.79253430005078459046597179521, −6.56707188979173338674783248161, −6.03608714094064893431400892804, −5.85005395599561985619280843546, −5.56485505107817768026900609829, −5.52212033311335163036353834290, −5.31659199757292068096333569047, −5.15063957064894969698554384189, −4.95880589178186559231077555954, −4.73570518297217134361075641773, −4.08372290730380335181854380122, −3.95350283387045037418919886907, −3.91982692896788961672814128614, −3.88451374751306410474508577850, −3.34417556426098123028691884245, −3.12731312007437915747506060590, −2.70665487370864572860153316653, −2.60554095707743777150431627647, −2.03334467768135975053386425005, −1.81790461288747954917250439259, −1.71510285824784588129406347453, −1.65312673379399869202513228298, −0.54594894991251944640608802693, −0.12143691650617283800001316686, 0.12143691650617283800001316686, 0.54594894991251944640608802693, 1.65312673379399869202513228298, 1.71510285824784588129406347453, 1.81790461288747954917250439259, 2.03334467768135975053386425005, 2.60554095707743777150431627647, 2.70665487370864572860153316653, 3.12731312007437915747506060590, 3.34417556426098123028691884245, 3.88451374751306410474508577850, 3.91982692896788961672814128614, 3.95350283387045037418919886907, 4.08372290730380335181854380122, 4.73570518297217134361075641773, 4.95880589178186559231077555954, 5.15063957064894969698554384189, 5.31659199757292068096333569047, 5.52212033311335163036353834290, 5.56485505107817768026900609829, 5.85005395599561985619280843546, 6.03608714094064893431400892804, 6.56707188979173338674783248161, 6.79253430005078459046597179521, 6.83488907603964730613411195985

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