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Properties

Label 18-2339e9-2339.2338-c0e9-0-0
Degree $18$
Conductor $2.095\times 10^{30}$
Sign $1$
Analytic cond. $4.02426$
Root an. cond. $1.08042$
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-s + 9·4-s − 5-s − 11-s − 9·12-s − 13-s + 15-s + 45·16-s − 19-s − 9·20-s + 33-s + 39-s − 41-s − 9·44-s − 45·48-s + 9·49-s − 9·52-s − 53-s + 55-s + 57-s − 59-s + 9·60-s + 165·64-s + 65-s − 67-s − 71-s − 73-s + ⋯
L(s)  = 1  − 3-s + 9·4-s − 5-s − 11-s − 9·12-s − 13-s + 15-s + 45·16-s − 19-s − 9·20-s + 33-s + 39-s − 41-s − 9·44-s − 45·48-s + 9·49-s − 9·52-s − 53-s + 55-s + 57-s − 59-s + 9·60-s + 165·64-s + 65-s − 67-s − 71-s − 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2339^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2339^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2339^{9}\)
Sign: $1$
Analytic conductor: \(4.02426\)
Root analytic conductor: \(1.08042\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2339} (2338, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2339^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(17.03393702\)
\(L(\frac12)\) \(\approx\) \(17.03393702\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2339 \( 1+O(T) \)
good2 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
23 \( ( 1 - T )^{9}( 1 + T )^{9} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( ( 1 - T )^{9}( 1 + T )^{9} \)
37 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
97 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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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

−3.51686805570511521382048065779, −3.33890241536102076590500829042, −3.33565047924022663251950581017, −3.25766152144892686472031061656, −3.11095346656242477471118709093, −3.07462780715204783813260831203, −2.88634755918027469564556805267, −2.67659744825457391585673361345, −2.59941363662503778233423905292, −2.55328698072505779576037191005, −2.52526266896793812265454205144, −2.39109456210827573813562317423, −2.37345953529544673061611154845, −2.30897775229814230531731645382, −2.29580953369293360905303185460, −2.00422563671398090515743335570, −1.92434362727813296042655541769, −1.56883179345890388832480677486, −1.55430357200260928998884344270, −1.54519574469416105406663387862, −1.30593907170785751700493778702, −1.26066619482374736120787607549, −1.19629690165450321804320148074, −0.847038502471603742596165660689, −0.72349883976655783718396239623, 0.72349883976655783718396239623, 0.847038502471603742596165660689, 1.19629690165450321804320148074, 1.26066619482374736120787607549, 1.30593907170785751700493778702, 1.54519574469416105406663387862, 1.55430357200260928998884344270, 1.56883179345890388832480677486, 1.92434362727813296042655541769, 2.00422563671398090515743335570, 2.29580953369293360905303185460, 2.30897775229814230531731645382, 2.37345953529544673061611154845, 2.39109456210827573813562317423, 2.52526266896793812265454205144, 2.55328698072505779576037191005, 2.59941363662503778233423905292, 2.67659744825457391585673361345, 2.88634755918027469564556805267, 3.07462780715204783813260831203, 3.11095346656242477471118709093, 3.25766152144892686472031061656, 3.33565047924022663251950581017, 3.33890241536102076590500829042, 3.51686805570511521382048065779

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

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