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  − 0.171·2-s − 1.97·4-s + 0.133·5-s + 0.615·7-s + 0.681·8-s − 0.0228·10-s − 11-s − 4.70·13-s − 0.105·14-s + 3.82·16-s − 2.34·17-s − 7.85·19-s − 0.262·20-s + 0.171·22-s − 1.86·23-s − 4.98·25-s + 0.807·26-s − 1.21·28-s − 4.53·29-s + 3.06·31-s − 2.01·32-s + 0.401·34-s + 0.0818·35-s + 9.24·37-s + 1.34·38-s + 0.0906·40-s − 1.52·41-s + ⋯
L(s)  = 1  − 0.121·2-s − 0.985·4-s + 0.0595·5-s + 0.232·7-s + 0.240·8-s − 0.00722·10-s − 0.301·11-s − 1.30·13-s − 0.0282·14-s + 0.956·16-s − 0.568·17-s − 1.80·19-s − 0.0586·20-s + 0.0365·22-s − 0.389·23-s − 0.996·25-s + 0.158·26-s − 0.229·28-s − 0.842·29-s + 0.551·31-s − 0.356·32-s + 0.0689·34-s + 0.0138·35-s + 1.51·37-s + 0.218·38-s + 0.0143·40-s − 0.238·41-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

\(L(1)\) \(\approx\) \(0.6988511154\)
\(L(\frac12)\) \(\approx\) \(0.6988511154\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.171T + 2T^{2} \)
5 \( 1 - 0.133T + 5T^{2} \)
7 \( 1 - 0.615T + 7T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 + 2.34T + 17T^{2} \)
19 \( 1 + 7.85T + 19T^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + 4.53T + 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 - 9.24T + 37T^{2} \)
41 \( 1 + 1.52T + 41T^{2} \)
43 \( 1 - 1.17T + 43T^{2} \)
47 \( 1 - 7.78T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 6.59T + 71T^{2} \)
73 \( 1 - 4.36T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−8.076440167438598036941164521449, −7.60497940364199476070425751861, −6.65583250085892292003807019731, −5.85246410088023014020733666386, −5.12389241134405138764927865614, −4.35324525760826573960085840302, −3.97276768538448586399276116703, −2.63062818652231044982840859196, −1.92539575037766351137464208310, −0.43224267904928466605158701461, 0.43224267904928466605158701461, 1.92539575037766351137464208310, 2.63062818652231044982840859196, 3.97276768538448586399276116703, 4.35324525760826573960085840302, 5.12389241134405138764927865614, 5.85246410088023014020733666386, 6.65583250085892292003807019731, 7.60497940364199476070425751861, 8.076440167438598036941164521449

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