Properties

Label 2-6039-1.1-c1-0-40
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  + 0.702·2-s − 1.50·4-s − 2.41·5-s + 0.380·7-s − 2.46·8-s − 1.69·10-s + 11-s + 2.87·13-s + 0.267·14-s + 1.28·16-s + 1.85·17-s − 2.38·19-s + 3.63·20-s + 0.702·22-s − 5.66·23-s + 0.811·25-s + 2.01·26-s − 0.573·28-s + 2.60·29-s − 4.67·31-s + 5.82·32-s + 1.30·34-s − 0.917·35-s − 10.7·37-s − 1.67·38-s + 5.93·40-s − 0.412·41-s + ⋯
L(s)  = 1  + 0.496·2-s − 0.753·4-s − 1.07·5-s + 0.143·7-s − 0.871·8-s − 0.535·10-s + 0.301·11-s + 0.796·13-s + 0.0714·14-s + 0.320·16-s + 0.449·17-s − 0.547·19-s + 0.811·20-s + 0.149·22-s − 1.18·23-s + 0.162·25-s + 0.396·26-s − 0.108·28-s + 0.483·29-s − 0.840·31-s + 1.03·32-s + 0.223·34-s − 0.155·35-s − 1.77·37-s − 0.272·38-s + 0.939·40-s − 0.0644·41-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

\(L(1)\) \(\approx\) \(1.147383594\)
\(L(\frac12)\) \(\approx\) \(1.147383594\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.702T + 2T^{2} \)
5 \( 1 + 2.41T + 5T^{2} \)
7 \( 1 - 0.380T + 7T^{2} \)
13 \( 1 - 2.87T + 13T^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
19 \( 1 + 2.38T + 19T^{2} \)
23 \( 1 + 5.66T + 23T^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 + 4.67T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 0.412T + 41T^{2} \)
43 \( 1 + 2.52T + 43T^{2} \)
47 \( 1 - 7.69T + 47T^{2} \)
53 \( 1 - 3.19T + 53T^{2} \)
59 \( 1 - 2.95T + 59T^{2} \)
67 \( 1 - 8.09T + 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 - 6.74T + 97T^{2} \)
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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

−8.206086535154702371778552268694, −7.45770392840612334214438951471, −6.56930125735301914984570591350, −5.82654815437674548105567404525, −5.11569197480808407984153001787, −4.24602776328073767912737091199, −3.79316875034273049117567609648, −3.22167679657598500593694251708, −1.83121768819141626945028923268, −0.52207521251493647622584291866, 0.52207521251493647622584291866, 1.83121768819141626945028923268, 3.22167679657598500593694251708, 3.79316875034273049117567609648, 4.24602776328073767912737091199, 5.11569197480808407984153001787, 5.82654815437674548105567404525, 6.56930125735301914984570591350, 7.45770392840612334214438951471, 8.206086535154702371778552268694

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