Properties

Label 2-6039-1.1-c1-0-248
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.26·2-s + 3.14·4-s + 2.83·5-s − 4.72·7-s + 2.60·8-s + 6.43·10-s + 11-s − 0.329·13-s − 10.7·14-s − 0.388·16-s − 6.43·17-s − 6.41·19-s + 8.92·20-s + 2.26·22-s − 0.000241·23-s + 3.03·25-s − 0.747·26-s − 14.8·28-s − 8.79·29-s + 5.66·31-s − 6.08·32-s − 14.6·34-s − 13.4·35-s − 0.385·37-s − 14.5·38-s + 7.37·40-s − 4.58·41-s + ⋯
L(s)  = 1  + 1.60·2-s + 1.57·4-s + 1.26·5-s − 1.78·7-s + 0.920·8-s + 2.03·10-s + 0.301·11-s − 0.0913·13-s − 2.86·14-s − 0.0971·16-s − 1.56·17-s − 1.47·19-s + 1.99·20-s + 0.483·22-s − 5.04e − 5·23-s + 0.607·25-s − 0.146·26-s − 2.81·28-s − 1.63·29-s + 1.01·31-s − 1.07·32-s − 2.50·34-s − 2.26·35-s − 0.0633·37-s − 2.36·38-s + 1.16·40-s − 0.716·41-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: \(1\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.26T + 2T^{2} \)
5 \( 1 - 2.83T + 5T^{2} \)
7 \( 1 + 4.72T + 7T^{2} \)
13 \( 1 + 0.329T + 13T^{2} \)
17 \( 1 + 6.43T + 17T^{2} \)
19 \( 1 + 6.41T + 19T^{2} \)
23 \( 1 + 0.000241T + 23T^{2} \)
29 \( 1 + 8.79T + 29T^{2} \)
31 \( 1 - 5.66T + 31T^{2} \)
37 \( 1 + 0.385T + 37T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 + 2.75T + 43T^{2} \)
47 \( 1 - 4.95T + 47T^{2} \)
53 \( 1 - 2.98T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
67 \( 1 + 7.49T + 67T^{2} \)
71 \( 1 - 3.18T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 5.72T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 3.95T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.14118824355598133816180620233, −6.55279749931437910883911515338, −6.23412172965852055578348243860, −5.72784324494550788902917238370, −4.78644199354832402238942314038, −4.05572224993485090511795253874, −3.35295279090300127765568011835, −2.46306062118150653482242738237, −1.98844833206833517194483469456, 0, 1.98844833206833517194483469456, 2.46306062118150653482242738237, 3.35295279090300127765568011835, 4.05572224993485090511795253874, 4.78644199354832402238942314038, 5.72784324494550788902917238370, 6.23412172965852055578348243860, 6.55279749931437910883911515338, 7.14118824355598133816180620233

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