Properties

Label 2-6039-1.1-c1-0-176
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.228·2-s − 1.94·4-s + 0.872·5-s − 1.27·7-s − 0.902·8-s + 0.199·10-s + 11-s + 5.51·13-s − 0.291·14-s + 3.68·16-s − 7.05·17-s − 4.53·19-s − 1.69·20-s + 0.228·22-s + 4.97·23-s − 4.23·25-s + 1.26·26-s + 2.48·28-s − 4.43·29-s + 6.49·31-s + 2.64·32-s − 1.61·34-s − 1.11·35-s + 4.65·37-s − 1.03·38-s − 0.787·40-s + 2.47·41-s + ⋯
L(s)  = 1  + 0.161·2-s − 0.973·4-s + 0.390·5-s − 0.481·7-s − 0.319·8-s + 0.0630·10-s + 0.301·11-s + 1.53·13-s − 0.0779·14-s + 0.922·16-s − 1.71·17-s − 1.04·19-s − 0.379·20-s + 0.0487·22-s + 1.03·23-s − 0.847·25-s + 0.247·26-s + 0.469·28-s − 0.822·29-s + 1.16·31-s + 0.468·32-s − 0.276·34-s − 0.187·35-s + 0.765·37-s − 0.168·38-s − 0.124·40-s + 0.387·41-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 \( 1 - 0.228T + 2T^{2} \)
5 \( 1 - 0.872T + 5T^{2} \)
7 \( 1 + 1.27T + 7T^{2} \)
13 \( 1 - 5.51T + 13T^{2} \)
17 \( 1 + 7.05T + 17T^{2} \)
19 \( 1 + 4.53T + 19T^{2} \)
23 \( 1 - 4.97T + 23T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
31 \( 1 - 6.49T + 31T^{2} \)
37 \( 1 - 4.65T + 37T^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 + 7.76T + 43T^{2} \)
47 \( 1 - 0.452T + 47T^{2} \)
53 \( 1 - 6.10T + 53T^{2} \)
59 \( 1 + 0.139T + 59T^{2} \)
67 \( 1 - 3.75T + 67T^{2} \)
71 \( 1 + 9.37T + 71T^{2} \)
73 \( 1 + 1.78T + 73T^{2} \)
79 \( 1 - 2.21T + 79T^{2} \)
83 \( 1 - 8.77T + 83T^{2} \)
89 \( 1 - 1.14T + 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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

−7.914737260421962366597254490108, −6.70614833933490003680225006628, −6.32243612055852883503089346471, −5.65070334851251958357579564267, −4.67599100641666562061585245719, −4.09981451305808155648213162506, −3.42290946756661328216989523993, −2.35825002580477014379236508490, −1.23654102542227728655752344015, 0, 1.23654102542227728655752344015, 2.35825002580477014379236508490, 3.42290946756661328216989523993, 4.09981451305808155648213162506, 4.67599100641666562061585245719, 5.65070334851251958357579564267, 6.32243612055852883503089346471, 6.70614833933490003680225006628, 7.914737260421962366597254490108

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