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Properties

Label 10-1167e5-1167.1166-c0e5-0-1
Degree $10$
Conductor $2.164\times 10^{15}$
Sign $1$
Analytic cond. $0.0670100$
Root an. cond. $0.763157$
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 + 5·3-s − 5·6-s − 7-s + 15·9-s − 13-s + 14-s − 15·18-s − 19-s − 5·21-s − 23-s + 5·25-s + 26-s + 35·27-s − 29-s + 38-s − 5·39-s + 5·42-s + 46-s − 47-s − 5·50-s − 53-s − 35·54-s − 5·57-s + 58-s − 15·63-s − 67-s + ⋯
L(s)  = 1  − 2-s + 5·3-s − 5·6-s − 7-s + 15·9-s − 13-s + 14-s − 15·18-s − 19-s − 5·21-s − 23-s + 5·25-s + 26-s + 35·27-s − 29-s + 38-s − 5·39-s + 5·42-s + 46-s − 47-s − 5·50-s − 53-s − 35·54-s − 5·57-s + 58-s − 15·63-s − 67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 389^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 389^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 389^{5}\)
Sign: $1$
Analytic conductor: \(0.0670100\)
Root analytic conductor: \(0.763157\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1167} (1166, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 389^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.287860277\)
\(L(\frac12)\) \(\approx\) \(3.287860277\)
\(L(1)\) 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)$
bad3$C_1$ \( ( 1 - T )^{5} \)
389$C_1$ \( ( 1 - T )^{5} \)
good2$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
7$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
13$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
19$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
23$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
29$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
47$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
53$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
67$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
71$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
73$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
79$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
83$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
89$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
97$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.36846682446945730190917947903, −6.09764420825306209542906928979, −5.79142176760021913483742394559, −5.72671745839240702413507683403, −5.21142217061171932491056936476, −4.83990771535483440572895408671, −4.82919926705836646398097860415, −4.69300577906107586637978379646, −4.52108157339005378501958274171, −4.47838611621930626202250587171, −4.01405220341645790386076565001, −3.83030355696749228678122164890, −3.81845821311401375802259956900, −3.47833707622938345388114532159, −3.14124393247825712580728090305, −3.09944494433352542573417133142, −2.99343319349850598639301025727, −2.82755539355373165210588171710, −2.55297543235675198126211238207, −2.31103571594715577029507389910, −2.24177598155137720708621661188, −1.85773581145050548503592024974, −1.43082190595531194856574421626, −1.26297138692205762875389068277, −1.21490124784255945794302989093, 1.21490124784255945794302989093, 1.26297138692205762875389068277, 1.43082190595531194856574421626, 1.85773581145050548503592024974, 2.24177598155137720708621661188, 2.31103571594715577029507389910, 2.55297543235675198126211238207, 2.82755539355373165210588171710, 2.99343319349850598639301025727, 3.09944494433352542573417133142, 3.14124393247825712580728090305, 3.47833707622938345388114532159, 3.81845821311401375802259956900, 3.83030355696749228678122164890, 4.01405220341645790386076565001, 4.47838611621930626202250587171, 4.52108157339005378501958274171, 4.69300577906107586637978379646, 4.82919926705836646398097860415, 4.83990771535483440572895408671, 5.21142217061171932491056936476, 5.72671745839240702413507683403, 5.79142176760021913483742394559, 6.09764420825306209542906928979, 6.36846682446945730190917947903

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