Properties

Label 2-6039-1.1-c1-0-61
Degree $2$
Conductor $6039$
Sign $1$
Analytic cond. $48.2216$
Root an. cond. $6.94418$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.04·2-s − 0.900·4-s + 4.37·5-s − 3.70·7-s + 3.04·8-s − 4.58·10-s − 11-s − 2.73·13-s + 3.88·14-s − 1.38·16-s + 3.57·17-s − 3.41·19-s − 3.93·20-s + 1.04·22-s + 3.58·23-s + 14.1·25-s + 2.86·26-s + 3.33·28-s + 1.76·29-s + 0.137·31-s − 4.62·32-s − 3.74·34-s − 16.2·35-s − 3.66·37-s + 3.58·38-s + 13.2·40-s − 1.23·41-s + ⋯
L(s)  = 1  − 0.741·2-s − 0.450·4-s + 1.95·5-s − 1.40·7-s + 1.07·8-s − 1.44·10-s − 0.301·11-s − 0.758·13-s + 1.03·14-s − 0.347·16-s + 0.866·17-s − 0.783·19-s − 0.880·20-s + 0.223·22-s + 0.747·23-s + 2.82·25-s + 0.562·26-s + 0.630·28-s + 0.327·29-s + 0.0247·31-s − 0.817·32-s − 0.642·34-s − 2.73·35-s − 0.601·37-s + 0.581·38-s + 2.10·40-s − 0.192·41-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: $1$
Analytic conductor: \(48.2216\)
Root analytic conductor: \(6.94418\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.238736530\)
\(L(\frac12)\) \(\approx\) \(1.238736530\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.04T + 2T^{2} \)
5 \( 1 - 4.37T + 5T^{2} \)
7 \( 1 + 3.70T + 7T^{2} \)
13 \( 1 + 2.73T + 13T^{2} \)
17 \( 1 - 3.57T + 17T^{2} \)
19 \( 1 + 3.41T + 19T^{2} \)
23 \( 1 - 3.58T + 23T^{2} \)
29 \( 1 - 1.76T + 29T^{2} \)
31 \( 1 - 0.137T + 31T^{2} \)
37 \( 1 + 3.66T + 37T^{2} \)
41 \( 1 + 1.23T + 41T^{2} \)
43 \( 1 + 5.20T + 43T^{2} \)
47 \( 1 + 1.30T + 47T^{2} \)
53 \( 1 - 6.66T + 53T^{2} \)
59 \( 1 - 8.84T + 59T^{2} \)
67 \( 1 - 0.443T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 6.52T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 4.85T + 83T^{2} \)
89 \( 1 + 6.77T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−8.423719216825307458724837444686, −7.18810907561694180350031250448, −6.78490540856550157294034736078, −5.92946356312571691587101797155, −5.35515482675546539456826596099, −4.65948331745823528797654412904, −3.40854149866124228792995752228, −2.60685628861368968300551125203, −1.75659421136802716908502376432, −0.65922018938889865794484106532, 0.65922018938889865794484106532, 1.75659421136802716908502376432, 2.60685628861368968300551125203, 3.40854149866124228792995752228, 4.65948331745823528797654412904, 5.35515482675546539456826596099, 5.92946356312571691587101797155, 6.78490540856550157294034736078, 7.18810907561694180350031250448, 8.423719216825307458724837444686

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