Properties

Label 18-311e9-311.310-c0e9-0-0
Degree $18$
Conductor $2.722\times 10^{22}$
Sign $1$
Analytic cond. $5.22706\times 10^{-8}$
Root an. cond. $0.393966$
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-s − 5-s + 6-s − 7-s + 10-s − 13-s + 14-s + 15-s + 21-s + 26-s − 30-s + 35-s + 39-s − 42-s − 47-s − 53-s + 65-s − 67-s − 70-s − 73-s − 78-s − 79-s − 83-s − 89-s + 91-s + 94-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s − 7-s + 10-s − 13-s + 14-s + 15-s + 21-s + 26-s − 30-s + 35-s + 39-s − 42-s − 47-s − 53-s + 65-s − 67-s − 70-s − 73-s − 78-s − 79-s − 83-s − 89-s + 91-s + 94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(311^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(311^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(311^{9}\)
Sign: $1$
Analytic conductor: \(5.22706\times 10^{-8}\)
Root analytic conductor: \(0.393966\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{311} (310, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 311^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02179064123\)
\(L(\frac12)\) \(\approx\) \(0.02179064123\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( ( 1 - T )^{9} \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
5 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
11 \( ( 1 - T )^{9}( 1 + T )^{9} \)
13 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( ( 1 - T )^{9}( 1 + T )^{9} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( ( 1 - T )^{9}( 1 + T )^{9} \)
37 \( ( 1 - T )^{9}( 1 + T )^{9} \)
41 \( ( 1 - T )^{9}( 1 + T )^{9} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
53 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
59 \( ( 1 - T )^{9}( 1 + T )^{9} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
67 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
71 \( ( 1 - T )^{9}( 1 + T )^{9} \)
73 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
79 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
83 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
89 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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

−5.08596653283961404526363527204, −4.93152808776806990433004663149, −4.87045767862458100823842747394, −4.76802420046159633678175499280, −4.76612353714826433759533398280, −4.47423513101626124125468636909, −4.32284630387690692803560108336, −4.15845217988444627872446609100, −3.97834659830279727603540882869, −3.82743606500904250131007296252, −3.78948720297622689490319437278, −3.55978987481579442972776591764, −3.55622150031991897712418300109, −3.24933682432361535870536268712, −3.02199171041639524893921194912, −2.95157743638267158317954348028, −2.76623714009693837417836326316, −2.61036875294852242430384586519, −2.60754925523510650193456179302, −2.25307333382695046017431241547, −1.92333721567853132709020595560, −1.66048916946803951063600971395, −1.55027124103382126593608965290, −1.40522464489028156044885062508, −0.68936555457351763988937341805, 0.68936555457351763988937341805, 1.40522464489028156044885062508, 1.55027124103382126593608965290, 1.66048916946803951063600971395, 1.92333721567853132709020595560, 2.25307333382695046017431241547, 2.60754925523510650193456179302, 2.61036875294852242430384586519, 2.76623714009693837417836326316, 2.95157743638267158317954348028, 3.02199171041639524893921194912, 3.24933682432361535870536268712, 3.55622150031991897712418300109, 3.55978987481579442972776591764, 3.78948720297622689490319437278, 3.82743606500904250131007296252, 3.97834659830279727603540882869, 4.15845217988444627872446609100, 4.32284630387690692803560108336, 4.47423513101626124125468636909, 4.76612353714826433759533398280, 4.76802420046159633678175499280, 4.87045767862458100823842747394, 4.93152808776806990433004663149, 5.08596653283961404526363527204

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

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