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Properties

Label 18-1671e9-1671.1670-c0e9-0-0
Degree $18$
Conductor $1.016\times 10^{29}$
Sign $1$
Analytic cond. $0.195075$
Root an. cond. $0.913201$
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 − 9·3-s + 5-s − 9·6-s − 7-s + 45·9-s + 10-s + 11-s − 14-s − 9·15-s + 45·18-s − 19-s + 9·21-s + 22-s + 23-s − 165·27-s − 9·30-s − 9·33-s − 35-s − 38-s + 41-s + 9·42-s − 43-s + 45·45-s + 46-s + 47-s + 53-s + ⋯
L(s)  = 1  + 2-s − 9·3-s + 5-s − 9·6-s − 7-s + 45·9-s + 10-s + 11-s − 14-s − 9·15-s + 45·18-s − 19-s + 9·21-s + 22-s + 23-s − 165·27-s − 9·30-s − 9·33-s − 35-s − 38-s + 41-s + 9·42-s − 43-s + 45·45-s + 46-s + 47-s + 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 557^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 557^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(3^{9} \cdot 557^{9}\)
Sign: $1$
Analytic conductor: \(0.195075\)
Root analytic conductor: \(0.913201\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1671} (1670, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 3^{9} \cdot 557^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05876328271\)
\(L(\frac12)\) \(\approx\) \(0.05876328271\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{9} \)
557 \( ( 1 + T )^{9} \)
good2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
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 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
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 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( ( 1 - T )^{9}( 1 + T )^{9} \)
37 \( ( 1 - T )^{9}( 1 + T )^{9} \)
41 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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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

−4.00869535216336956319935601644, −3.88801000521698560103234006393, −3.88135750065285215189030247080, −3.83278602319135456493431987092, −3.67459827134676097920089417540, −3.66117185652938655987884260349, −3.20572420508609169572972314011, −3.04494633285290967538426472302, −3.01013153347103660235309929914, −2.98000694871258629850290422272, −2.77298342848437064096016472808, −2.32235608023896616934057633168, −2.27860037132295437384101162753, −2.04244874444611431063975806443, −2.04014194154852307593356486324, −1.76851513362048831706608994184, −1.72958095942354866392015631615, −1.64871969398023675169113668130, −1.51717725844726943945485803639, −1.21243269892397948615240873489, −1.11298873160910866547329218775, −0.977323044950498296553514312375, −0.894943822898025597571574450536, −0.57502123343045510622356873323, −0.35493752612622000602068088784, 0.35493752612622000602068088784, 0.57502123343045510622356873323, 0.894943822898025597571574450536, 0.977323044950498296553514312375, 1.11298873160910866547329218775, 1.21243269892397948615240873489, 1.51717725844726943945485803639, 1.64871969398023675169113668130, 1.72958095942354866392015631615, 1.76851513362048831706608994184, 2.04014194154852307593356486324, 2.04244874444611431063975806443, 2.27860037132295437384101162753, 2.32235608023896616934057633168, 2.77298342848437064096016472808, 2.98000694871258629850290422272, 3.01013153347103660235309929914, 3.04494633285290967538426472302, 3.20572420508609169572972314011, 3.66117185652938655987884260349, 3.67459827134676097920089417540, 3.83278602319135456493431987092, 3.88135750065285215189030247080, 3.88801000521698560103234006393, 4.00869535216336956319935601644

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

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