Properties

Label 2-6039-1.1-c1-0-209
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.53·2-s + 0.360·4-s − 2.16·5-s + 2.96·7-s − 2.51·8-s − 3.32·10-s + 11-s + 1.11·13-s + 4.55·14-s − 4.59·16-s − 6.92·17-s + 6.93·19-s − 0.780·20-s + 1.53·22-s + 0.602·23-s − 0.303·25-s + 1.71·26-s + 1.06·28-s − 4.99·29-s − 1.25·31-s − 2.01·32-s − 10.6·34-s − 6.42·35-s − 3.94·37-s + 10.6·38-s + 5.45·40-s + 1.36·41-s + ⋯
L(s)  = 1  + 1.08·2-s + 0.180·4-s − 0.969·5-s + 1.12·7-s − 0.890·8-s − 1.05·10-s + 0.301·11-s + 0.310·13-s + 1.21·14-s − 1.14·16-s − 1.67·17-s + 1.59·19-s − 0.174·20-s + 0.327·22-s + 0.125·23-s − 0.0606·25-s + 0.336·26-s + 0.201·28-s − 0.926·29-s − 0.224·31-s − 0.356·32-s − 1.82·34-s − 1.08·35-s − 0.648·37-s + 1.72·38-s + 0.863·40-s + 0.213·41-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: \(1\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.53T + 2T^{2} \)
5 \( 1 + 2.16T + 5T^{2} \)
7 \( 1 - 2.96T + 7T^{2} \)
13 \( 1 - 1.11T + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 - 6.93T + 19T^{2} \)
23 \( 1 - 0.602T + 23T^{2} \)
29 \( 1 + 4.99T + 29T^{2} \)
31 \( 1 + 1.25T + 31T^{2} \)
37 \( 1 + 3.94T + 37T^{2} \)
41 \( 1 - 1.36T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 3.48T + 47T^{2} \)
53 \( 1 + 9.97T + 53T^{2} \)
59 \( 1 + 4.08T + 59T^{2} \)
67 \( 1 + 5.56T + 67T^{2} \)
71 \( 1 + 6.01T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 3.58T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−7.57334588046699844737973274727, −7.07922293460276345850298316174, −6.05984358058107506027207406730, −5.41484138019693614197961444886, −4.62758583600298502452296631542, −4.19753755458656201582033643276, −3.51946475956294723072889219644, −2.60038319175842598232822593336, −1.43974673249606540165431078647, 0, 1.43974673249606540165431078647, 2.60038319175842598232822593336, 3.51946475956294723072889219644, 4.19753755458656201582033643276, 4.62758583600298502452296631542, 5.41484138019693614197961444886, 6.05984358058107506027207406730, 7.07922293460276345850298316174, 7.57334588046699844737973274727

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