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Properties

Label 18-3071e9-3071.3070-c0e9-0-0
Degree $18$
Conductor $2.430\times 10^{31}$
Sign $1$
Analytic cond. $46.6586$
Root an. cond. $1.23799$
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 − 7-s + 10-s − 11-s − 13-s + 14-s + 15-s − 19-s + 21-s + 22-s + 26-s − 30-s + 33-s + 35-s + 9·37-s + 38-s + 39-s − 41-s − 42-s − 43-s + 55-s + 57-s + 65-s − 66-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s − 7-s + 10-s − 11-s − 13-s + 14-s + 15-s − 19-s + 21-s + 22-s + 26-s − 30-s + 33-s + 35-s + 9·37-s + 38-s + 39-s − 41-s − 42-s − 43-s + 55-s + 57-s + 65-s − 66-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{9} \cdot 83^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{9} \cdot 83^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(37^{9} \cdot 83^{9}\)
Sign: $1$
Analytic conductor: \(46.6586\)
Root analytic conductor: \(1.23799\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3071} (3070, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 37^{9} \cdot 83^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1870731648\)
\(L(\frac12)\) \(\approx\) \(0.1870731648\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( ( 1 - T )^{9} \)
83 \( ( 1 - T )^{9} \)
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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
53 \( ( 1 - T )^{9}( 1 + T )^{9} \)
59 \( ( 1 - T )^{9}( 1 + T )^{9} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
67 \( ( 1 - T )^{9}( 1 + T )^{9} \)
71 \( ( 1 - T )^{9}( 1 + T )^{9} \)
73 \( ( 1 - T )^{9}( 1 + T )^{9} \)
79 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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

−3.47422603321756887312079677806, −3.42085736204238470352840313540, −3.36135142280380622193723895406, −3.27408903523905709847585045290, −3.21303874876934899385232004195, −2.84880041864042457047760529977, −2.76016420742714548981832216133, −2.70509341142104894586605063840, −2.51242618462220176185006634405, −2.47230874685646000895357440553, −2.43582974170144929043405417210, −2.42554026079841362336165161746, −2.38783159635550580122095103327, −2.22512685540384958979203254013, −2.00529257537597998204374337251, −1.82526515188482448438553975553, −1.52216391020012590796902112342, −1.46499724702278981410264685253, −1.45897012478896017274208130692, −1.23736374340032259533330928724, −0.846573950580842810361645784506, −0.840139402366781433680078391020, −0.794704674331274182283567024818, −0.42330758725496727446582694991, −0.34949458421228608669766997738, 0.34949458421228608669766997738, 0.42330758725496727446582694991, 0.794704674331274182283567024818, 0.840139402366781433680078391020, 0.846573950580842810361645784506, 1.23736374340032259533330928724, 1.45897012478896017274208130692, 1.46499724702278981410264685253, 1.52216391020012590796902112342, 1.82526515188482448438553975553, 2.00529257537597998204374337251, 2.22512685540384958979203254013, 2.38783159635550580122095103327, 2.42554026079841362336165161746, 2.43582974170144929043405417210, 2.47230874685646000895357440553, 2.51242618462220176185006634405, 2.70509341142104894586605063840, 2.76016420742714548981832216133, 2.84880041864042457047760529977, 3.21303874876934899385232004195, 3.27408903523905709847585045290, 3.36135142280380622193723895406, 3.42085736204238470352840313540, 3.47422603321756887312079677806

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

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