Properties

Label 2-6039-1.1-c1-0-10
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  − 1.92·2-s + 1.68·4-s − 2.68·5-s − 3.03·7-s + 0.597·8-s + 5.15·10-s − 11-s − 1.87·13-s + 5.82·14-s − 4.52·16-s + 5.85·17-s + 2.10·19-s − 4.53·20-s + 1.92·22-s − 5.06·23-s + 2.21·25-s + 3.59·26-s − 5.12·28-s − 0.755·29-s − 7.72·31-s + 7.49·32-s − 11.2·34-s + 8.14·35-s − 8.48·37-s − 4.04·38-s − 1.60·40-s + 7.29·41-s + ⋯
L(s)  = 1  − 1.35·2-s + 0.844·4-s − 1.20·5-s − 1.14·7-s + 0.211·8-s + 1.63·10-s − 0.301·11-s − 0.519·13-s + 1.55·14-s − 1.13·16-s + 1.42·17-s + 0.482·19-s − 1.01·20-s + 0.409·22-s − 1.05·23-s + 0.442·25-s + 0.705·26-s − 0.968·28-s − 0.140·29-s − 1.38·31-s + 1.32·32-s − 1.92·34-s + 1.37·35-s − 1.39·37-s − 0.655·38-s − 0.253·40-s + 1.13·41-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

\(L(1)\) \(\approx\) \(0.1650156414\)
\(L(\frac12)\) \(\approx\) \(0.1650156414\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.92T + 2T^{2} \)
5 \( 1 + 2.68T + 5T^{2} \)
7 \( 1 + 3.03T + 7T^{2} \)
13 \( 1 + 1.87T + 13T^{2} \)
17 \( 1 - 5.85T + 17T^{2} \)
19 \( 1 - 2.10T + 19T^{2} \)
23 \( 1 + 5.06T + 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 + 7.72T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 7.29T + 41T^{2} \)
43 \( 1 - 2.41T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
67 \( 1 + 7.27T + 67T^{2} \)
71 \( 1 - 9.06T + 71T^{2} \)
73 \( 1 - 0.597T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 4.45T + 89T^{2} \)
97 \( 1 + 0.962T + 97T^{2} \)
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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

−7.962371745907575317272908143981, −7.52221057069193049729838621241, −7.17444564272508480222223582684, −6.12718983147980648519474029680, −5.32563624653540328353263661526, −4.22553387784704878584318044474, −3.55143150920350255049371675694, −2.74265326531615857887217042550, −1.48749424424555006208308026289, −0.26908594034325422804353611704, 0.26908594034325422804353611704, 1.48749424424555006208308026289, 2.74265326531615857887217042550, 3.55143150920350255049371675694, 4.22553387784704878584318044474, 5.32563624653540328353263661526, 6.12718983147980648519474029680, 7.17444564272508480222223582684, 7.52221057069193049729838621241, 7.962371745907575317272908143981

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