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  − 2.53·2-s + 4.44·4-s − 0.994·5-s + 4.88·7-s − 6.19·8-s + 2.52·10-s − 11-s + 4.29·13-s − 12.4·14-s + 6.83·16-s + 3.82·17-s − 2.93·19-s − 4.41·20-s + 2.53·22-s + 5.41·23-s − 4.01·25-s − 10.8·26-s + 21.6·28-s + 2.31·29-s − 6.92·31-s − 4.95·32-s − 9.70·34-s − 4.85·35-s + 0.949·37-s + 7.43·38-s + 6.15·40-s + 4.66·41-s + ⋯
L(s)  = 1  − 1.79·2-s + 2.22·4-s − 0.444·5-s + 1.84·7-s − 2.18·8-s + 0.797·10-s − 0.301·11-s + 1.18·13-s − 3.31·14-s + 1.70·16-s + 0.927·17-s − 0.672·19-s − 0.986·20-s + 0.541·22-s + 1.12·23-s − 0.802·25-s − 2.13·26-s + 4.10·28-s + 0.428·29-s − 1.24·31-s − 0.876·32-s − 1.66·34-s − 0.821·35-s + 0.156·37-s + 1.20·38-s + 0.973·40-s + 0.727·41-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

\(L(1)\) \(\approx\) \(1.089767037\)
\(L(\frac12)\) \(\approx\) \(1.089767037\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.53T + 2T^{2} \)
5 \( 1 + 0.994T + 5T^{2} \)
7 \( 1 - 4.88T + 7T^{2} \)
13 \( 1 - 4.29T + 13T^{2} \)
17 \( 1 - 3.82T + 17T^{2} \)
19 \( 1 + 2.93T + 19T^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 - 2.31T + 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 - 0.949T + 37T^{2} \)
41 \( 1 - 4.66T + 41T^{2} \)
43 \( 1 - 4.23T + 43T^{2} \)
47 \( 1 + 8.27T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 5.12T + 59T^{2} \)
67 \( 1 + 6.83T + 67T^{2} \)
71 \( 1 + 4.72T + 71T^{2} \)
73 \( 1 - 15.9T + 73T^{2} \)
79 \( 1 + 6.64T + 79T^{2} \)
83 \( 1 - 8.78T + 83T^{2} \)
89 \( 1 + 0.0660T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−7.966623448001541611244911388248, −7.80863916235191716858059682525, −7.11676130701291470563926217360, −6.13820844437226057584504629134, −5.38293629525553634593715258496, −4.43655843606963737738348980271, −3.44795189598041439200097831196, −2.27447924026645081866529319175, −1.52022422365320208498467372728, −0.78627884101900068742061038523, 0.78627884101900068742061038523, 1.52022422365320208498467372728, 2.27447924026645081866529319175, 3.44795189598041439200097831196, 4.43655843606963737738348980271, 5.38293629525553634593715258496, 6.13820844437226057584504629134, 7.11676130701291470563926217360, 7.80863916235191716858059682525, 7.966623448001541611244911388248

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