Properties

Label 2-6039-1.1-c1-0-225
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  + 2.62·2-s + 4.86·4-s + 4.12·5-s − 0.639·7-s + 7.51·8-s + 10.8·10-s + 11-s + 3.39·13-s − 1.67·14-s + 9.96·16-s + 2.27·17-s − 3.83·19-s + 20.0·20-s + 2.62·22-s + 0.231·23-s + 12.0·25-s + 8.90·26-s − 3.11·28-s − 5.21·29-s − 6.79·31-s + 11.0·32-s + 5.96·34-s − 2.63·35-s + 0.00835·37-s − 10.0·38-s + 31.0·40-s − 1.25·41-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.43·4-s + 1.84·5-s − 0.241·7-s + 2.65·8-s + 3.42·10-s + 0.301·11-s + 0.942·13-s − 0.447·14-s + 2.49·16-s + 0.551·17-s − 0.880·19-s + 4.49·20-s + 0.558·22-s + 0.0483·23-s + 2.40·25-s + 1.74·26-s − 0.588·28-s − 0.968·29-s − 1.21·31-s + 1.95·32-s + 1.02·34-s − 0.446·35-s + 0.00137·37-s − 1.63·38-s + 4.90·40-s − 0.195·41-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

\(L(1)\) \(\approx\) \(9.959321162\)
\(L(\frac12)\) \(\approx\) \(9.959321162\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.62T + 2T^{2} \)
5 \( 1 - 4.12T + 5T^{2} \)
7 \( 1 + 0.639T + 7T^{2} \)
13 \( 1 - 3.39T + 13T^{2} \)
17 \( 1 - 2.27T + 17T^{2} \)
19 \( 1 + 3.83T + 19T^{2} \)
23 \( 1 - 0.231T + 23T^{2} \)
29 \( 1 + 5.21T + 29T^{2} \)
31 \( 1 + 6.79T + 31T^{2} \)
37 \( 1 - 0.00835T + 37T^{2} \)
41 \( 1 + 1.25T + 41T^{2} \)
43 \( 1 + 5.91T + 43T^{2} \)
47 \( 1 + 2.04T + 47T^{2} \)
53 \( 1 + 5.78T + 53T^{2} \)
59 \( 1 + 6.98T + 59T^{2} \)
67 \( 1 + 4.12T + 67T^{2} \)
71 \( 1 - 2.69T + 71T^{2} \)
73 \( 1 - 6.26T + 73T^{2} \)
79 \( 1 - 3.96T + 79T^{2} \)
83 \( 1 - 2.54T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 + 5.19T + 97T^{2} \)
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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

−7.80717364913841238229681510579, −6.74543597855455921942300952002, −6.43489122722744435005216914131, −5.77952191934939808734698680977, −5.35840809269586933477986576121, −4.56128853405612433802801376542, −3.60830485266020831839074828278, −3.02113300917144431615993694352, −1.97057953054917317115755685055, −1.56358808313746662543533341604, 1.56358808313746662543533341604, 1.97057953054917317115755685055, 3.02113300917144431615993694352, 3.60830485266020831839074828278, 4.56128853405612433802801376542, 5.35840809269586933477986576121, 5.77952191934939808734698680977, 6.43489122722744435005216914131, 6.74543597855455921942300952002, 7.80717364913841238229681510579

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