Properties

Label 2-6039-1.1-c1-0-123
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  − 2.54·2-s + 4.46·4-s + 0.268·5-s + 2.27·7-s − 6.27·8-s − 0.682·10-s + 11-s + 2.39·13-s − 5.78·14-s + 7.03·16-s + 6.76·17-s + 3.70·19-s + 1.19·20-s − 2.54·22-s + 8.19·23-s − 4.92·25-s − 6.08·26-s + 10.1·28-s − 6.34·29-s + 8.61·31-s − 5.32·32-s − 17.2·34-s + 0.609·35-s + 7.70·37-s − 9.43·38-s − 1.68·40-s + 2.20·41-s + ⋯
L(s)  = 1  − 1.79·2-s + 2.23·4-s + 0.119·5-s + 0.859·7-s − 2.21·8-s − 0.215·10-s + 0.301·11-s + 0.663·13-s − 1.54·14-s + 1.75·16-s + 1.64·17-s + 0.851·19-s + 0.268·20-s − 0.542·22-s + 1.70·23-s − 0.985·25-s − 1.19·26-s + 1.92·28-s − 1.17·29-s + 1.54·31-s − 0.941·32-s − 2.95·34-s + 0.103·35-s + 1.26·37-s − 1.53·38-s − 0.266·40-s + 0.345·41-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

\(L(1)\) \(\approx\) \(1.309540076\)
\(L(\frac12)\) \(\approx\) \(1.309540076\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.54T + 2T^{2} \)
5 \( 1 - 0.268T + 5T^{2} \)
7 \( 1 - 2.27T + 7T^{2} \)
13 \( 1 - 2.39T + 13T^{2} \)
17 \( 1 - 6.76T + 17T^{2} \)
19 \( 1 - 3.70T + 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 + 6.34T + 29T^{2} \)
31 \( 1 - 8.61T + 31T^{2} \)
37 \( 1 - 7.70T + 37T^{2} \)
41 \( 1 - 2.20T + 41T^{2} \)
43 \( 1 - 5.10T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 7.23T + 59T^{2} \)
67 \( 1 - 0.912T + 67T^{2} \)
71 \( 1 + 9.89T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 - 7.24T + 79T^{2} \)
83 \( 1 + 5.85T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−8.110124545850662382544026995211, −7.53120198440436120997517070487, −7.15906734608578008006152238273, −6.01349726732227944409479490760, −5.59586415228133592146475639803, −4.42140661777193592475386120913, −3.29613919052178244532181988757, −2.45848553839438829971140786494, −1.29632229559408550294035536398, −0.967198034930396598429626354649, 0.967198034930396598429626354649, 1.29632229559408550294035536398, 2.45848553839438829971140786494, 3.29613919052178244532181988757, 4.42140661777193592475386120913, 5.59586415228133592146475639803, 6.01349726732227944409479490760, 7.15906734608578008006152238273, 7.53120198440436120997517070487, 8.110124545850662382544026995211

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