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  − 0.822·2-s − 1.32·4-s − 2.11·5-s + 0.404·7-s + 2.73·8-s + 1.74·10-s − 11-s + 4.06·13-s − 0.332·14-s + 0.398·16-s − 0.0205·17-s + 3.97·19-s + 2.80·20-s + 0.822·22-s + 8.67·23-s − 0.522·25-s − 3.34·26-s − 0.534·28-s + 5.89·29-s + 5.40·31-s − 5.79·32-s + 0.0168·34-s − 0.855·35-s + 1.99·37-s − 3.27·38-s − 5.78·40-s − 6.26·41-s + ⋯
L(s)  = 1  − 0.581·2-s − 0.661·4-s − 0.946·5-s + 0.152·7-s + 0.966·8-s + 0.550·10-s − 0.301·11-s + 1.12·13-s − 0.0888·14-s + 0.0996·16-s − 0.00498·17-s + 0.912·19-s + 0.626·20-s + 0.175·22-s + 1.80·23-s − 0.104·25-s − 0.655·26-s − 0.101·28-s + 1.09·29-s + 0.970·31-s − 1.02·32-s + 0.00289·34-s − 0.144·35-s + 0.328·37-s − 0.530·38-s − 0.914·40-s − 0.979·41-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

\(L(1)\) \(\approx\) \(1.051578540\)
\(L(\frac12)\) \(\approx\) \(1.051578540\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.822T + 2T^{2} \)
5 \( 1 + 2.11T + 5T^{2} \)
7 \( 1 - 0.404T + 7T^{2} \)
13 \( 1 - 4.06T + 13T^{2} \)
17 \( 1 + 0.0205T + 17T^{2} \)
19 \( 1 - 3.97T + 19T^{2} \)
23 \( 1 - 8.67T + 23T^{2} \)
29 \( 1 - 5.89T + 29T^{2} \)
31 \( 1 - 5.40T + 31T^{2} \)
37 \( 1 - 1.99T + 37T^{2} \)
41 \( 1 + 6.26T + 41T^{2} \)
43 \( 1 - 7.16T + 43T^{2} \)
47 \( 1 + 8.53T + 47T^{2} \)
53 \( 1 + 5.18T + 53T^{2} \)
59 \( 1 - 0.638T + 59T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 4.60T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 0.603T + 97T^{2} \)
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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

−8.244988716371284465577913382270, −7.57199506332408438568150268995, −6.89781514556665772966091584509, −5.96868159621879896010088393025, −4.95026022286658819776213683210, −4.56579692053916840201372873802, −3.58637915340869932605615191214, −2.99262585711353404462263204532, −1.42015122165833679228723903519, −0.66351049452631776571306192733, 0.66351049452631776571306192733, 1.42015122165833679228723903519, 2.99262585711353404462263204532, 3.58637915340869932605615191214, 4.56579692053916840201372873802, 4.95026022286658819776213683210, 5.96868159621879896010088393025, 6.89781514556665772966091584509, 7.57199506332408438568150268995, 8.244988716371284465577913382270

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