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  + 1.33·2-s − 0.228·4-s + 2.36·5-s − 2.92·7-s − 2.96·8-s + 3.15·10-s − 11-s + 4.67·13-s − 3.88·14-s − 3.49·16-s + 3.08·17-s − 5.01·19-s − 0.541·20-s − 1.33·22-s + 4.73·23-s + 0.608·25-s + 6.21·26-s + 0.668·28-s + 2.14·29-s + 1.22·31-s + 1.28·32-s + 4.10·34-s − 6.91·35-s − 5.71·37-s − 6.68·38-s − 7.02·40-s + 9.30·41-s + ⋯
L(s)  = 1  + 0.941·2-s − 0.114·4-s + 1.05·5-s − 1.10·7-s − 1.04·8-s + 0.996·10-s − 0.301·11-s + 1.29·13-s − 1.03·14-s − 0.872·16-s + 0.748·17-s − 1.15·19-s − 0.121·20-s − 0.283·22-s + 0.986·23-s + 0.121·25-s + 1.21·26-s + 0.126·28-s + 0.398·29-s + 0.220·31-s + 0.227·32-s + 0.704·34-s − 1.16·35-s − 0.940·37-s − 1.08·38-s − 1.11·40-s + 1.45·41-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

\(L(1)\) \(\approx\) \(2.909237952\)
\(L(\frac12)\) \(\approx\) \(2.909237952\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.33T + 2T^{2} \)
5 \( 1 - 2.36T + 5T^{2} \)
7 \( 1 + 2.92T + 7T^{2} \)
13 \( 1 - 4.67T + 13T^{2} \)
17 \( 1 - 3.08T + 17T^{2} \)
19 \( 1 + 5.01T + 19T^{2} \)
23 \( 1 - 4.73T + 23T^{2} \)
29 \( 1 - 2.14T + 29T^{2} \)
31 \( 1 - 1.22T + 31T^{2} \)
37 \( 1 + 5.71T + 37T^{2} \)
41 \( 1 - 9.30T + 41T^{2} \)
43 \( 1 + 3.00T + 43T^{2} \)
47 \( 1 - 1.19T + 47T^{2} \)
53 \( 1 - 4.66T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
67 \( 1 - 8.99T + 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 0.425T + 89T^{2} \)
97 \( 1 - 5.28T + 97T^{2} \)
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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

−8.239379319707403119928881295402, −6.98783091505813199661268501094, −6.36345721006109907752354468928, −5.88206289137453767902452497127, −5.35967212024127005886737845772, −4.42623529069720064424971529653, −3.60331134661001512193059127158, −3.03246540844220197134004029878, −2.11793738331804301058112119051, −0.77113869456304046969718285908, 0.77113869456304046969718285908, 2.11793738331804301058112119051, 3.03246540844220197134004029878, 3.60331134661001512193059127158, 4.42623529069720064424971529653, 5.35967212024127005886737845772, 5.88206289137453767902452497127, 6.36345721006109907752354468928, 6.98783091505813199661268501094, 8.239379319707403119928881295402

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