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  − 2.73·2-s + 5.50·4-s − 0.604·5-s − 1.91·7-s − 9.59·8-s + 1.65·10-s − 11-s + 0.176·13-s + 5.23·14-s + 15.2·16-s − 2.74·17-s + 3.64·19-s − 3.32·20-s + 2.73·22-s − 1.61·23-s − 4.63·25-s − 0.484·26-s − 10.5·28-s + 7.55·29-s + 7.88·31-s − 22.6·32-s + 7.51·34-s + 1.15·35-s + 8.43·37-s − 9.99·38-s + 5.80·40-s − 8.93·41-s + ⋯
L(s)  = 1  − 1.93·2-s + 2.75·4-s − 0.270·5-s − 0.723·7-s − 3.39·8-s + 0.523·10-s − 0.301·11-s + 0.0490·13-s + 1.40·14-s + 3.81·16-s − 0.665·17-s + 0.836·19-s − 0.743·20-s + 0.583·22-s − 0.337·23-s − 0.926·25-s − 0.0950·26-s − 1.98·28-s + 1.40·29-s + 1.41·31-s − 4.00·32-s + 1.28·34-s + 0.195·35-s + 1.38·37-s − 1.62·38-s + 0.917·40-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: $1$
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

\(L(1)\) \(\approx\) \(0.4611173876\)
\(L(\frac12)\) \(\approx\) \(0.4611173876\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 2T^{2} \)
5 \( 1 + 0.604T + 5T^{2} \)
7 \( 1 + 1.91T + 7T^{2} \)
13 \( 1 - 0.176T + 13T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
19 \( 1 - 3.64T + 19T^{2} \)
23 \( 1 + 1.61T + 23T^{2} \)
29 \( 1 - 7.55T + 29T^{2} \)
31 \( 1 - 7.88T + 31T^{2} \)
37 \( 1 - 8.43T + 37T^{2} \)
41 \( 1 + 8.93T + 41T^{2} \)
43 \( 1 + 4.51T + 43T^{2} \)
47 \( 1 - 2.48T + 47T^{2} \)
53 \( 1 - 7.57T + 53T^{2} \)
59 \( 1 + 1.33T + 59T^{2} \)
67 \( 1 + 5.50T + 67T^{2} \)
71 \( 1 + 0.330T + 71T^{2} \)
73 \( 1 + 1.75T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 4.11T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−8.246270308773443110179817650388, −7.58054054319750593920528601609, −6.83525470354589475786653398615, −6.37191384216618337561357013836, −5.59875096294225284641537230042, −4.32764885482710282374246937298, −3.11890413398886700135267037131, −2.62308719815214231552673592790, −1.51667631295249208967555247252, −0.48463816212071062368203895018, 0.48463816212071062368203895018, 1.51667631295249208967555247252, 2.62308719815214231552673592790, 3.11890413398886700135267037131, 4.32764885482710282374246937298, 5.59875096294225284641537230042, 6.37191384216618337561357013836, 6.83525470354589475786653398615, 7.58054054319750593920528601609, 8.246270308773443110179817650388

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