Properties

Label 18-1759e9-1759.1758-c0e9-0-0
Degree $18$
Conductor $1.612\times 10^{29}$
Sign $1$
Analytic cond. $0.309604$
Root an. cond. $0.936939$
Motivic weight $0$
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  − 3·8-s + 9·9-s + 9·49-s + 3·64-s − 27·72-s + 45·81-s − 9·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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  − 3·8-s + 9·9-s + 9·49-s + 3·64-s − 27·72-s + 45·81-s − 9·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1759^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1759^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(1759^{9}\)
Sign: $1$
Analytic conductor: \(0.309604\)
Root analytic conductor: \(0.936939\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1759} (1758, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 1759^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.468914233\)
\(L(\frac12)\) \(\approx\) \(3.468914233\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1759 \( ( 1 - T )^{9} \)
good2 \( ( 1 + T^{3} + T^{6} )^{3} \)
3 \( ( 1 - T )^{9}( 1 + T )^{9} \)
5 \( 1 + T^{9} + T^{18} \)
7 \( ( 1 - T )^{9}( 1 + T )^{9} \)
11 \( 1 + T^{9} + T^{18} \)
13 \( 1 + T^{9} + T^{18} \)
17 \( 1 + T^{9} + T^{18} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( 1 + T^{9} + T^{18} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( ( 1 + T^{3} + T^{6} )^{3} \)
37 \( ( 1 - T )^{9}( 1 + T )^{9} \)
41 \( 1 + T^{9} + T^{18} \)
43 \( 1 + T^{9} + T^{18} \)
47 \( 1 + T^{9} + T^{18} \)
53 \( 1 + T^{9} + T^{18} \)
59 \( ( 1 - T )^{9}( 1 + T )^{9} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
67 \( ( 1 - T )^{9}( 1 + T )^{9} \)
71 \( 1 + T^{9} + T^{18} \)
73 \( ( 1 - T )^{9}( 1 + T )^{9} \)
79 \( ( 1 - T )^{9}( 1 + T )^{9} \)
83 \( ( 1 - T )^{9}( 1 + T )^{9} \)
89 \( ( 1 + T + T^{2} )^{9} \)
97 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.77928689416932821339181011048, −3.76180282040575080016932319009, −3.74325151887482480483162447904, −3.59263129206123631210820425569, −3.41585794429366520961547592743, −3.38025648270400829781420620684, −3.37649185220190720502312086484, −2.83808812683631469716646747113, −2.83544565113122554110013057887, −2.82567915544817036974319969874, −2.73272695669211770281982213339, −2.42497148044393525087466020339, −2.40266769330274586588337832141, −2.26072148890558489440652546343, −2.15669021924031906706072368420, −2.05217548609731020577389200686, −1.83064701045503901792890330934, −1.79938216648024758676482222416, −1.45591807625374347847185314761, −1.42037068983218769544812341960, −1.19581152659483251335862479881, −1.17620689604478159479229350130, −1.04558241225060459297725624437, −0.968888773396792202436930200982, −0.75367348543036938401868719099, 0.75367348543036938401868719099, 0.968888773396792202436930200982, 1.04558241225060459297725624437, 1.17620689604478159479229350130, 1.19581152659483251335862479881, 1.42037068983218769544812341960, 1.45591807625374347847185314761, 1.79938216648024758676482222416, 1.83064701045503901792890330934, 2.05217548609731020577389200686, 2.15669021924031906706072368420, 2.26072148890558489440652546343, 2.40266769330274586588337832141, 2.42497148044393525087466020339, 2.73272695669211770281982213339, 2.82567915544817036974319969874, 2.83544565113122554110013057887, 2.83808812683631469716646747113, 3.37649185220190720502312086484, 3.38025648270400829781420620684, 3.41585794429366520961547592743, 3.59263129206123631210820425569, 3.74325151887482480483162447904, 3.76180282040575080016932319009, 3.77928689416932821339181011048

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

Plot not available for L-functions of degree greater than 10.