Properties

Degree $2$
Conductor $6039$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.805·2-s − 1.35·4-s − 1.06·5-s − 0.203·7-s − 2.69·8-s − 0.857·10-s − 11-s + 0.801·13-s − 0.163·14-s + 0.530·16-s + 0.376·17-s + 2.62·19-s + 1.43·20-s − 0.805·22-s − 2.93·23-s − 3.86·25-s + 0.645·26-s + 0.274·28-s − 0.907·29-s + 2.07·31-s + 5.82·32-s + 0.303·34-s + 0.216·35-s − 2.14·37-s + 2.11·38-s + 2.87·40-s + 1.83·41-s + ⋯
L(s)  = 1  + 0.569·2-s − 0.675·4-s − 0.476·5-s − 0.0767·7-s − 0.954·8-s − 0.271·10-s − 0.301·11-s + 0.222·13-s − 0.0436·14-s + 0.132·16-s + 0.0913·17-s + 0.601·19-s + 0.321·20-s − 0.171·22-s − 0.611·23-s − 0.773·25-s + 0.126·26-s + 0.0518·28-s − 0.168·29-s + 0.373·31-s + 1.02·32-s + 0.0519·34-s + 0.0365·35-s − 0.353·37-s + 0.342·38-s + 0.454·40-s + 0.286·41-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: $1$
Motivic weight: \(1\)
Character: $\chi_{6039} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.310138744\)
\(L(\frac12)\) \(\approx\) \(1.310138744\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.805T + 2T^{2} \)
5 \( 1 + 1.06T + 5T^{2} \)
7 \( 1 + 0.203T + 7T^{2} \)
13 \( 1 - 0.801T + 13T^{2} \)
17 \( 1 - 0.376T + 17T^{2} \)
19 \( 1 - 2.62T + 19T^{2} \)
23 \( 1 + 2.93T + 23T^{2} \)
29 \( 1 + 0.907T + 29T^{2} \)
31 \( 1 - 2.07T + 31T^{2} \)
37 \( 1 + 2.14T + 37T^{2} \)
41 \( 1 - 1.83T + 41T^{2} \)
43 \( 1 + 2.25T + 43T^{2} \)
47 \( 1 - 0.685T + 47T^{2} \)
53 \( 1 + 14.0T + 53T^{2} \)
59 \( 1 + 7.73T + 59T^{2} \)
67 \( 1 + 7.59T + 67T^{2} \)
71 \( 1 - 9.23T + 71T^{2} \)
73 \( 1 - 7.49T + 73T^{2} \)
79 \( 1 - 7.41T + 79T^{2} \)
83 \( 1 - 8.55T + 83T^{2} \)
89 \( 1 - 2.97T + 89T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−7.970809546572430854663519717600, −7.57154886773991522889436370247, −6.40036934645768394018839938727, −5.92872124080197756718523410941, −5.00784406469487804977830653740, −4.54062673643027605180679481604, −3.58092773307889068767588234064, −3.19640891384361402164014782629, −1.92293049102032235174648026791, −0.54295278966997822666812862566, 0.54295278966997822666812862566, 1.92293049102032235174648026791, 3.19640891384361402164014782629, 3.58092773307889068767588234064, 4.54062673643027605180679481604, 5.00784406469487804977830653740, 5.92872124080197756718523410941, 6.40036934645768394018839938727, 7.57154886773991522889436370247, 7.970809546572430854663519717600

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