Properties

Degree $2$
Conductor $6039$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.50·2-s + 0.252·4-s + 0.569·5-s − 3.97·7-s + 2.62·8-s − 0.855·10-s − 11-s − 3.48·13-s + 5.96·14-s − 4.44·16-s − 7.75·17-s + 1.56·19-s + 0.143·20-s + 1.50·22-s − 4.91·23-s − 4.67·25-s + 5.23·26-s − 1.00·28-s − 6.29·29-s + 2.87·31-s + 1.41·32-s + 11.6·34-s − 2.26·35-s − 9.34·37-s − 2.34·38-s + 1.49·40-s − 1.58·41-s + ⋯
L(s)  = 1  − 1.06·2-s + 0.126·4-s + 0.254·5-s − 1.50·7-s + 0.927·8-s − 0.270·10-s − 0.301·11-s − 0.967·13-s + 1.59·14-s − 1.11·16-s − 1.88·17-s + 0.357·19-s + 0.0321·20-s + 0.319·22-s − 1.02·23-s − 0.935·25-s + 1.02·26-s − 0.189·28-s − 1.16·29-s + 0.516·31-s + 0.250·32-s + 1.99·34-s − 0.382·35-s − 1.53·37-s − 0.379·38-s + 0.236·40-s − 0.247·41-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: $1$
Motivic weight: \(1\)
Character: $\chi_{6039} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05282325951\)
\(L(\frac12)\) \(\approx\) \(0.05282325951\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.50T + 2T^{2} \)
5 \( 1 - 0.569T + 5T^{2} \)
7 \( 1 + 3.97T + 7T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 + 7.75T + 17T^{2} \)
19 \( 1 - 1.56T + 19T^{2} \)
23 \( 1 + 4.91T + 23T^{2} \)
29 \( 1 + 6.29T + 29T^{2} \)
31 \( 1 - 2.87T + 31T^{2} \)
37 \( 1 + 9.34T + 37T^{2} \)
41 \( 1 + 1.58T + 41T^{2} \)
43 \( 1 - 5.31T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 0.418T + 59T^{2} \)
67 \( 1 - 9.84T + 67T^{2} \)
71 \( 1 + 9.93T + 71T^{2} \)
73 \( 1 - 7.11T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 4.04T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + 1.77T + 97T^{2} \)
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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

−8.120673128324600300123503064302, −7.45648463985113840739682262908, −6.79669467804188882929042412721, −6.19733733879651175601385555773, −5.25771454834155621807961662635, −4.39410113597000615451264521439, −3.60207029141601916212570425013, −2.50348011103919694704074360061, −1.80919977223428335133748038831, −0.13395458072333980008735686771, 0.13395458072333980008735686771, 1.80919977223428335133748038831, 2.50348011103919694704074360061, 3.60207029141601916212570425013, 4.39410113597000615451264521439, 5.25771454834155621807961662635, 6.19733733879651175601385555773, 6.79669467804188882929042412721, 7.45648463985113840739682262908, 8.120673128324600300123503064302

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