Properties

Label 8-1400e4-1.1-c0e4-0-2
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $0.238309$
Root an. cond. $0.835877$
Motivic weight $0$
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  − 2-s − 3·3-s + 5-s + 3·6-s + 4·7-s + 6·9-s − 10-s + 2·13-s − 4·14-s − 3·15-s − 6·18-s + 2·19-s − 12·21-s − 2·23-s − 2·26-s − 10·27-s + 3·30-s + 32-s + 4·35-s − 2·38-s − 6·39-s + 12·42-s + 6·45-s + 2·46-s + 10·49-s + 10·54-s − 6·57-s + ⋯
L(s)  = 1  − 2-s − 3·3-s + 5-s + 3·6-s + 4·7-s + 6·9-s − 10-s + 2·13-s − 4·14-s − 3·15-s − 6·18-s + 2·19-s − 12·21-s − 2·23-s − 2·26-s − 10·27-s + 3·30-s + 32-s + 4·35-s − 2·38-s − 6·39-s + 12·42-s + 6·45-s + 2·46-s + 10·49-s + 10·54-s − 6·57-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.238309\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5360146764\)
\(L(\frac12)\) \(\approx\) \(0.5360146764\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
good3$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
23$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
61$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
67$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
71$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
89$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−7.12171918416976001221394753408, −6.63469588782169186030921665953, −6.43520394811542215432462433999, −6.25137562299408477533209363567, −6.18801518191244997682381618804, −5.87374430171563989399850384813, −5.69763042689353483256138357909, −5.54470601327486007760926593254, −5.42355156779689226948832990600, −5.06861614393463777531848420907, −4.90216729498878839613961551833, −4.63179191282320981710034938650, −4.38394864580764029181426475342, −4.37348790438039167033285866280, −4.35538968792354698104366005420, −3.53212327050473597218943166149, −3.49296635226964067475845602398, −3.34593507378023861084359138157, −2.30713421645267976560776689669, −2.14507414811926742479397129886, −2.00286324226684587564999816918, −1.43905388031223434604736478736, −1.36166945667014705849167415532, −1.25102279053126631379284794267, −0.856756275800495936277887822617, 0.856756275800495936277887822617, 1.25102279053126631379284794267, 1.36166945667014705849167415532, 1.43905388031223434604736478736, 2.00286324226684587564999816918, 2.14507414811926742479397129886, 2.30713421645267976560776689669, 3.34593507378023861084359138157, 3.49296635226964067475845602398, 3.53212327050473597218943166149, 4.35538968792354698104366005420, 4.37348790438039167033285866280, 4.38394864580764029181426475342, 4.63179191282320981710034938650, 4.90216729498878839613961551833, 5.06861614393463777531848420907, 5.42355156779689226948832990600, 5.54470601327486007760926593254, 5.69763042689353483256138357909, 5.87374430171563989399850384813, 6.18801518191244997682381618804, 6.25137562299408477533209363567, 6.43520394811542215432462433999, 6.63469588782169186030921665953, 7.12171918416976001221394753408

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