Properties

Label 2-6039-1.1-c1-0-213
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  + 2.31·2-s + 3.37·4-s + 2.47·5-s + 3.26·7-s + 3.18·8-s + 5.74·10-s − 11-s + 6.05·13-s + 7.57·14-s + 0.629·16-s + 6.63·17-s − 4.68·19-s + 8.36·20-s − 2.31·22-s + 6.72·23-s + 1.14·25-s + 14.0·26-s + 11.0·28-s + 6.44·29-s − 6.17·31-s − 4.90·32-s + 15.3·34-s + 8.10·35-s − 6.59·37-s − 10.8·38-s + 7.88·40-s − 8.91·41-s + ⋯
L(s)  = 1  + 1.63·2-s + 1.68·4-s + 1.10·5-s + 1.23·7-s + 1.12·8-s + 1.81·10-s − 0.301·11-s + 1.67·13-s + 2.02·14-s + 0.157·16-s + 1.60·17-s − 1.07·19-s + 1.86·20-s − 0.494·22-s + 1.40·23-s + 0.229·25-s + 2.75·26-s + 2.08·28-s + 1.19·29-s − 1.10·31-s − 0.866·32-s + 2.63·34-s + 1.36·35-s − 1.08·37-s − 1.76·38-s + 1.24·40-s − 1.39·41-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

\(L(1)\) \(\approx\) \(8.351468420\)
\(L(\frac12)\) \(\approx\) \(8.351468420\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.31T + 2T^{2} \)
5 \( 1 - 2.47T + 5T^{2} \)
7 \( 1 - 3.26T + 7T^{2} \)
13 \( 1 - 6.05T + 13T^{2} \)
17 \( 1 - 6.63T + 17T^{2} \)
19 \( 1 + 4.68T + 19T^{2} \)
23 \( 1 - 6.72T + 23T^{2} \)
29 \( 1 - 6.44T + 29T^{2} \)
31 \( 1 + 6.17T + 31T^{2} \)
37 \( 1 + 6.59T + 37T^{2} \)
41 \( 1 + 8.91T + 41T^{2} \)
43 \( 1 + 8.98T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 3.81T + 59T^{2} \)
67 \( 1 - 3.27T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 6.29T + 73T^{2} \)
79 \( 1 + 5.89T + 79T^{2} \)
83 \( 1 - 6.77T + 83T^{2} \)
89 \( 1 - 2.86T + 89T^{2} \)
97 \( 1 - 8.49T + 97T^{2} \)
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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

−8.141297084148019575800445696226, −6.96371941847619912476651060403, −6.38322410985918222677903866740, −5.74319102851425006933875444487, −5.08554736276790698289724758322, −4.76706278075950709325733061310, −3.55469113046909570511853946334, −3.14346096574963001617403191717, −1.85834950938306942531490538669, −1.46687867678198896744910628507, 1.46687867678198896744910628507, 1.85834950938306942531490538669, 3.14346096574963001617403191717, 3.55469113046909570511853946334, 4.76706278075950709325733061310, 5.08554736276790698289724758322, 5.74319102851425006933875444487, 6.38322410985918222677903866740, 6.96371941847619912476651060403, 8.141297084148019575800445696226

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