Properties

Label 2-6039-1.1-c1-0-221
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  + 2.22·2-s + 2.97·4-s − 3.25·5-s − 0.940·7-s + 2.16·8-s − 7.26·10-s + 11-s + 1.71·13-s − 2.09·14-s − 1.11·16-s + 0.0942·17-s + 3.20·19-s − 9.67·20-s + 2.22·22-s + 6.07·23-s + 5.60·25-s + 3.83·26-s − 2.79·28-s + 3.73·29-s − 9.85·31-s − 6.81·32-s + 0.210·34-s + 3.06·35-s − 8.79·37-s + 7.15·38-s − 7.04·40-s − 3.45·41-s + ⋯
L(s)  = 1  + 1.57·2-s + 1.48·4-s − 1.45·5-s − 0.355·7-s + 0.764·8-s − 2.29·10-s + 0.301·11-s + 0.476·13-s − 0.560·14-s − 0.279·16-s + 0.0228·17-s + 0.736·19-s − 2.16·20-s + 0.475·22-s + 1.26·23-s + 1.12·25-s + 0.751·26-s − 0.527·28-s + 0.694·29-s − 1.76·31-s − 1.20·32-s + 0.0360·34-s + 0.517·35-s − 1.44·37-s + 1.16·38-s − 1.11·40-s − 0.539·41-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 \( 1 - 2.22T + 2T^{2} \)
5 \( 1 + 3.25T + 5T^{2} \)
7 \( 1 + 0.940T + 7T^{2} \)
13 \( 1 - 1.71T + 13T^{2} \)
17 \( 1 - 0.0942T + 17T^{2} \)
19 \( 1 - 3.20T + 19T^{2} \)
23 \( 1 - 6.07T + 23T^{2} \)
29 \( 1 - 3.73T + 29T^{2} \)
31 \( 1 + 9.85T + 31T^{2} \)
37 \( 1 + 8.79T + 37T^{2} \)
41 \( 1 + 3.45T + 41T^{2} \)
43 \( 1 + 9.13T + 43T^{2} \)
47 \( 1 - 4.55T + 47T^{2} \)
53 \( 1 + 2.28T + 53T^{2} \)
59 \( 1 + 0.677T + 59T^{2} \)
67 \( 1 + 8.66T + 67T^{2} \)
71 \( 1 + 7.54T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 4.70T + 79T^{2} \)
83 \( 1 + 0.268T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 19.6T + 97T^{2} \)
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

−7.23472029480489271252660103543, −7.08079962857633133153256602400, −6.20398367077282929650537430837, −5.31503124740200586305300822268, −4.79256785490664110623926173749, −3.94210548321619905855715015364, −3.42206897792854004097831340938, −2.96797177600219518048702970611, −1.52450807616281701110377801710, 0, 1.52450807616281701110377801710, 2.96797177600219518048702970611, 3.42206897792854004097831340938, 3.94210548321619905855715015364, 4.79256785490664110623926173749, 5.31503124740200586305300822268, 6.20398367077282929650537430837, 7.08079962857633133153256602400, 7.23472029480489271252660103543

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