Properties

Label 8-160e4-1.1-c1e4-0-1
Degree $8$
Conductor $655360000$
Sign $1$
Analytic cond. $2.66433$
Root an. cond. $1.13031$
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  + 10·25-s + 24·29-s + 2·81-s − 24·89-s − 72·101-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 2·25-s + 4.45·29-s + 2/9·81-s − 2.54·89-s − 7.16·101-s − 4·121-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 + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2.66433\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{160} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.457780710\)
\(L(\frac12)\) \(\approx\) \(1.457780710\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
47$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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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

−9.376028053791290275288534376888, −9.300931496355539349042987080767, −8.931235541128439524930930999071, −8.439039713800605599546754603685, −8.348622798880760706844654204351, −8.290531743834660130877881971853, −8.023666956296543312925465170693, −7.42541760646773038739521629905, −7.24428262020010073229159694421, −6.74347725118726718629208840651, −6.65989392962173684406985324141, −6.43519184526866561268525595499, −6.31002618409748606745822405705, −5.43352400992124616358714228380, −5.33588751391846393555097535154, −5.27231567091486983092416864152, −4.58001151796818634463342727799, −4.32262036432335171751972215288, −4.24641408342865867467795153238, −3.60042315231287594743817683746, −2.86185531446723818518215656358, −2.75334185645567540086841645499, −2.71110513399110985174563244987, −1.49656336476859230117027825145, −1.05253035425592649295608201792, 1.05253035425592649295608201792, 1.49656336476859230117027825145, 2.71110513399110985174563244987, 2.75334185645567540086841645499, 2.86185531446723818518215656358, 3.60042315231287594743817683746, 4.24641408342865867467795153238, 4.32262036432335171751972215288, 4.58001151796818634463342727799, 5.27231567091486983092416864152, 5.33588751391846393555097535154, 5.43352400992124616358714228380, 6.31002618409748606745822405705, 6.43519184526866561268525595499, 6.65989392962173684406985324141, 6.74347725118726718629208840651, 7.24428262020010073229159694421, 7.42541760646773038739521629905, 8.023666956296543312925465170693, 8.290531743834660130877881971853, 8.348622798880760706844654204351, 8.439039713800605599546754603685, 8.931235541128439524930930999071, 9.300931496355539349042987080767, 9.376028053791290275288534376888

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