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Properties

Label 18-2879e9-2879.2878-c0e9-0-0
Degree $18$
Conductor $1.359\times 10^{31}$
Sign $1$
Analytic cond. $26.0969$
Root an. cond. $1.19866$
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  − 2-s − 3-s − 5-s + 6-s + 10-s − 11-s + 15-s − 19-s + 22-s − 30-s − 31-s + 33-s − 37-s + 38-s − 41-s + 9·49-s − 53-s + 55-s + 57-s + 18·59-s − 61-s + 62-s − 66-s − 71-s − 73-s + 74-s − 79-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 10-s − 11-s + 15-s − 19-s + 22-s − 30-s − 31-s + 33-s − 37-s + 38-s − 41-s + 9·49-s − 53-s + 55-s + 57-s + 18·59-s − 61-s + 62-s − 66-s − 71-s − 73-s + 74-s − 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2879^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2879^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2879^{9}\)
Sign: $1$
Analytic conductor: \(26.0969\)
Root analytic conductor: \(1.19866\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2879} (2878, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2879^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4436303362\)
\(L(\frac12)\) \(\approx\) \(0.4436303362\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2879 \( 1+O(T) \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
11 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
13 \( ( 1 - T )^{9}( 1 + T )^{9} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
23 \( ( 1 - T )^{9}( 1 + T )^{9} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
37 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
41 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( ( 1 - T )^{18} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
97 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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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

−3.76448440305187982494114049914, −3.57363836361127876737240780833, −3.28958771209006820384142982294, −3.26888927978261531829473162275, −3.01580914669361114577159375545, −2.94825228310707758368943618711, −2.76021606848038630086910044951, −2.75852276672476276694301306589, −2.68165626245376032797686156058, −2.67189553005013359825617443294, −2.31346974639404210476414136037, −2.29193684961185127575040482204, −2.18584954972560792914021467817, −2.15785034511169637743536253718, −1.97677274933957989073447726935, −1.95454277490332421152351002401, −1.89344074553777657726297187868, −1.50225261777666614101810561839, −1.36481561393861522199650122463, −1.19697833933997140836522994464, −0.816338774889594519469691319869, −0.814320480915327338674438062149, −0.78089095362398695857548138602, −0.61196878332961710459581551533, −0.46095908825813691786870255938, 0.46095908825813691786870255938, 0.61196878332961710459581551533, 0.78089095362398695857548138602, 0.814320480915327338674438062149, 0.816338774889594519469691319869, 1.19697833933997140836522994464, 1.36481561393861522199650122463, 1.50225261777666614101810561839, 1.89344074553777657726297187868, 1.95454277490332421152351002401, 1.97677274933957989073447726935, 2.15785034511169637743536253718, 2.18584954972560792914021467817, 2.29193684961185127575040482204, 2.31346974639404210476414136037, 2.67189553005013359825617443294, 2.68165626245376032797686156058, 2.75852276672476276694301306589, 2.76021606848038630086910044951, 2.94825228310707758368943618711, 3.01580914669361114577159375545, 3.26888927978261531829473162275, 3.28958771209006820384142982294, 3.57363836361127876737240780833, 3.76448440305187982494114049914

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

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