Properties

Label 2-6039-1.1-c1-0-152
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  + 0.534·2-s − 1.71·4-s − 3.62·5-s + 4.31·7-s − 1.98·8-s − 1.93·10-s + 11-s − 1.21·13-s + 2.30·14-s + 2.36·16-s − 0.492·17-s + 0.909·19-s + 6.21·20-s + 0.534·22-s − 9.05·23-s + 8.14·25-s − 0.651·26-s − 7.39·28-s + 3.36·29-s − 1.80·31-s + 5.23·32-s − 0.263·34-s − 15.6·35-s − 2.52·37-s + 0.485·38-s + 7.19·40-s + 3.50·41-s + ⋯
L(s)  = 1  + 0.377·2-s − 0.857·4-s − 1.62·5-s + 1.62·7-s − 0.701·8-s − 0.612·10-s + 0.301·11-s − 0.338·13-s + 0.615·14-s + 0.592·16-s − 0.119·17-s + 0.208·19-s + 1.39·20-s + 0.113·22-s − 1.88·23-s + 1.62·25-s − 0.127·26-s − 1.39·28-s + 0.624·29-s − 0.323·31-s + 0.925·32-s − 0.0451·34-s − 2.64·35-s − 0.414·37-s + 0.0787·38-s + 1.13·40-s + 0.547·41-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 \( 1 - 0.534T + 2T^{2} \)
5 \( 1 + 3.62T + 5T^{2} \)
7 \( 1 - 4.31T + 7T^{2} \)
13 \( 1 + 1.21T + 13T^{2} \)
17 \( 1 + 0.492T + 17T^{2} \)
19 \( 1 - 0.909T + 19T^{2} \)
23 \( 1 + 9.05T + 23T^{2} \)
29 \( 1 - 3.36T + 29T^{2} \)
31 \( 1 + 1.80T + 31T^{2} \)
37 \( 1 + 2.52T + 37T^{2} \)
41 \( 1 - 3.50T + 41T^{2} \)
43 \( 1 - 1.17T + 43T^{2} \)
47 \( 1 - 6.47T + 47T^{2} \)
53 \( 1 + 7.40T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
67 \( 1 + 1.39T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + 9.92T + 73T^{2} \)
79 \( 1 - 4.62T + 79T^{2} \)
83 \( 1 - 2.80T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 - 2.52T + 97T^{2} \)
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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

−7.79625592107812477306183341068, −7.33417683044002266055635061071, −6.18680567632314050755750641440, −5.25872842487190860876686262902, −4.69696138871856031522078023013, −4.06738358156433451388588041710, −3.68321410838408512020511487417, −2.41689922938729550646049531448, −1.14510831013924597962284683978, 0, 1.14510831013924597962284683978, 2.41689922938729550646049531448, 3.68321410838408512020511487417, 4.06738358156433451388588041710, 4.69696138871856031522078023013, 5.25872842487190860876686262902, 6.18680567632314050755750641440, 7.33417683044002266055635061071, 7.79625592107812477306183341068

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