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Properties

Label 2-1539-19.18-c0-0-5
Degree $2$
Conductor $1539$
Sign $1$
Analytic cond. $0.768061$
Root an. cond. $0.876390$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 7-s + 1.73·11-s + 16-s − 1.73·17-s − 19-s − 1.73·23-s − 25-s + 28-s − 43-s + 1.73·44-s + 1.73·47-s + 61-s + 64-s − 1.73·68-s − 73-s − 76-s + 1.73·77-s − 1.73·92-s − 100-s − 1.73·101-s + 112-s − 1.73·119-s + ⋯
L(s)  = 1  + 4-s + 7-s + 1.73·11-s + 16-s − 1.73·17-s − 19-s − 1.73·23-s − 25-s + 28-s − 43-s + 1.73·44-s + 1.73·47-s + 61-s + 64-s − 1.73·68-s − 73-s − 76-s + 1.73·77-s − 1.73·92-s − 100-s − 1.73·101-s + 112-s − 1.73·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (892, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.548700859\)
\(L(\frac12)\) \(\approx\) \(1.548700859\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + T \)
good2 \( 1 - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.73T + T^{2} \)
23 \( 1 + 1.73T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.73T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.665810550497869689288904253595, −8.699305756940304096565234984531, −8.120895673236630269391164907949, −7.10190288565279087104627082370, −6.45657761209922086487914873689, −5.79965151018111122626072051208, −4.40296315023034032603407102348, −3.86608318904438766214677956294, −2.24477997958511877895802242719, −1.67079199837300348335689397064, 1.67079199837300348335689397064, 2.24477997958511877895802242719, 3.86608318904438766214677956294, 4.40296315023034032603407102348, 5.79965151018111122626072051208, 6.45657761209922086487914873689, 7.10190288565279087104627082370, 8.120895673236630269391164907949, 8.699305756940304096565234984531, 9.665810550497869689288904253595

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