Properties

Label 2-6039-1.1-c1-0-42
Degree $2$
Conductor $6039$
Sign $1$
Analytic cond. $48.2216$
Root an. cond. $6.94418$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.34·2-s + 3.49·4-s − 3.40·5-s − 2.95·7-s − 3.50·8-s + 7.97·10-s − 11-s + 5.60·13-s + 6.93·14-s + 1.23·16-s + 3.44·17-s + 6.93·19-s − 11.8·20-s + 2.34·22-s − 0.836·23-s + 6.57·25-s − 13.1·26-s − 10.3·28-s − 6.68·29-s + 0.117·31-s + 4.12·32-s − 8.06·34-s + 10.0·35-s − 0.152·37-s − 16.2·38-s + 11.9·40-s + 10.6·41-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.74·4-s − 1.52·5-s − 1.11·7-s − 1.24·8-s + 2.52·10-s − 0.301·11-s + 1.55·13-s + 1.85·14-s + 0.307·16-s + 0.834·17-s + 1.59·19-s − 2.65·20-s + 0.499·22-s − 0.174·23-s + 1.31·25-s − 2.57·26-s − 1.95·28-s − 1.24·29-s + 0.0210·31-s + 0.729·32-s − 1.38·34-s + 1.70·35-s − 0.0250·37-s − 2.63·38-s + 1.88·40-s + 1.65·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $1$
Analytic conductor: \(48.2216\)
Root analytic conductor: \(6.94418\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4766784966\)
\(L(\frac12)\) \(\approx\) \(0.4766784966\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
61 \( 1 + T \)
good2 \( 1 + 2.34T + 2T^{2} \)
5 \( 1 + 3.40T + 5T^{2} \)
7 \( 1 + 2.95T + 7T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 - 6.93T + 19T^{2} \)
23 \( 1 + 0.836T + 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 - 0.117T + 31T^{2} \)
37 \( 1 + 0.152T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 5.87T + 43T^{2} \)
47 \( 1 + 0.889T + 47T^{2} \)
53 \( 1 + 3.79T + 53T^{2} \)
59 \( 1 - 9.37T + 59T^{2} \)
67 \( 1 - 2.66T + 67T^{2} \)
71 \( 1 + 1.13T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 5.23T + 79T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 + 2.71T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−7.971551592064745040591862931467, −7.65391501146611511270239949739, −7.06990466945476431027034155115, −6.23305926118158289998042687388, −5.48120931433481931952382179396, −4.07467097462352269209128262697, −3.49226075868698767195524200594, −2.78313489646529542883149903226, −1.28931020500917756085555491963, −0.52789755650064710048276365649, 0.52789755650064710048276365649, 1.28931020500917756085555491963, 2.78313489646529542883149903226, 3.49226075868698767195524200594, 4.07467097462352269209128262697, 5.48120931433481931952382179396, 6.23305926118158289998042687388, 7.06990466945476431027034155115, 7.65391501146611511270239949739, 7.971551592064745040591862931467

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