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Properties

Label 18-3047e9-3047.3046-c0e9-0-0
Degree $18$
Conductor $2.264\times 10^{31}$
Sign $1$
Analytic cond. $43.4776$
Root an. cond. $1.23314$
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 − 6-s − 9·11-s + 17-s − 9·22-s − 23-s + 9·25-s + 9·33-s + 34-s + 43-s − 46-s − 47-s + 9·49-s + 9·50-s − 51-s − 59-s + 61-s + 9·66-s − 67-s + 69-s − 71-s + 73-s − 9·75-s + 86-s − 89-s − 94-s + ⋯
L(s)  = 1  + 2-s − 3-s − 6-s − 9·11-s + 17-s − 9·22-s − 23-s + 9·25-s + 9·33-s + 34-s + 43-s − 46-s − 47-s + 9·49-s + 9·50-s − 51-s − 59-s + 61-s + 9·66-s − 67-s + 69-s − 71-s + 73-s − 9·75-s + 86-s − 89-s − 94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{9} \cdot 277^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{9} \cdot 277^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(11^{9} \cdot 277^{9}\)
Sign: $1$
Analytic conductor: \(43.4776\)
Root analytic conductor: \(1.23314\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3047} (3046, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 11^{9} \cdot 277^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.581632450\)
\(L(\frac12)\) \(\approx\) \(1.581632450\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 + T )^{9} \)
277 \( ( 1 + T )^{9} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
7 \( ( 1 - T )^{9}( 1 + T )^{9} \)
13 \( ( 1 - T )^{9}( 1 + T )^{9} \)
17 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
59 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
61 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
83 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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

−3.33890368954899840179572652311, −3.33757573281242882899306859746, −3.25824487825193391052641518143, −3.22677998695739377443595983404, −2.97738270228729689162418196650, −2.86653334412630322810884534766, −2.83141237580480548926724994733, −2.82618369805585075805560949164, −2.69534716998685152058438628797, −2.68871794946399040821343610953, −2.65481955415069219422893959826, −2.33490716046622404510778119267, −2.28193081984813901012391377669, −2.20508275768796743089982919184, −2.12862305588353151514623288154, −1.95540761607969560942627481144, −1.92725057303957678325895373430, −1.67529401188552451410301656266, −1.24926736862978714687133868083, −1.23091352317974092917032863054, −0.987266740590171539537648420138, −0.68817995446352793428717389772, −0.66298361864021316371439684410, −0.63039617193672386831685274196, −0.61359183505421999188099443638, 0.61359183505421999188099443638, 0.63039617193672386831685274196, 0.66298361864021316371439684410, 0.68817995446352793428717389772, 0.987266740590171539537648420138, 1.23091352317974092917032863054, 1.24926736862978714687133868083, 1.67529401188552451410301656266, 1.92725057303957678325895373430, 1.95540761607969560942627481144, 2.12862305588353151514623288154, 2.20508275768796743089982919184, 2.28193081984813901012391377669, 2.33490716046622404510778119267, 2.65481955415069219422893959826, 2.68871794946399040821343610953, 2.69534716998685152058438628797, 2.82618369805585075805560949164, 2.83141237580480548926724994733, 2.86653334412630322810884534766, 2.97738270228729689162418196650, 3.22677998695739377443595983404, 3.25824487825193391052641518143, 3.33757573281242882899306859746, 3.33890368954899840179572652311

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

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