↑

Properties

Label 2-130-65.9-c1-0-4
Degree $2$
Conductor $130$
Sign $0.804 - 0.594i$
Analytic cond. $1.03805$
Root an. cond. $1.01884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (1.45 + 0.837i)3-s + (0.499 + 0.866i)4-s + (−1.67 − 1.48i)5-s + (0.837 + 1.45i)6-s + (1.56 − 0.903i)7-s + 0.999i·8-s + (−0.0969 − 0.167i)9-s + (−0.710 − 2.12i)10-s + (−3.22 + 5.58i)11-s + 1.67i·12-s + (−1.98 − 3.01i)13-s + 1.80·14-s + (−1.18 − 3.55i)15-s + (−0.5 + 0.866i)16-s + (0.416 − 0.240i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.837 + 0.483i)3-s + (0.249 + 0.433i)4-s + (−0.749 − 0.662i)5-s + (0.341 + 0.592i)6-s + (0.591 − 0.341i)7-s + 0.353i·8-s + (−0.0323 − 0.0559i)9-s + (−0.224 − 0.670i)10-s + (−0.971 + 1.68i)11-s + 0.483i·12-s + (−0.549 − 0.835i)13-s + 0.482·14-s + (−0.307 − 0.917i)15-s + (−0.125 + 0.216i)16-s + (0.101 − 0.0583i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.804 - 0.594i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.804 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55719 + 0.513083i\)
\(L(\frac12)\) \(\approx\) \(1.55719 + 0.513083i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.67 + 1.48i)T \)
13 \( 1 + (1.98 + 3.01i)T \)
good3 \( 1 + (-1.45 - 0.837i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.56 + 0.903i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.22 - 5.58i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.416 + 0.240i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.16 - 3.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.15 + 2.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.25T + 31T^{2} \)
37 \( 1 + (2.65 + 1.53i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.75 - 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.19iT - 47T^{2} \)
53 \( 1 - 0.906iT - 53T^{2} \)
59 \( 1 + (-3.28 - 5.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.47 - 9.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.562 + 0.324i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.83 + 3.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.60iT - 73T^{2} \)
79 \( 1 - 2.29T + 79T^{2} \)
83 \( 1 + 13.3iT - 83T^{2} \)
89 \( 1 + (0.578 - 1.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.9 + 6.91i)T + (48.5 - 84.0i)T^{2} \)
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

−13.34840206406733205074814998658, −12.67014459845153648591327270311, −11.58321491117248786353475935029, −10.25258644989697718510551459611, −9.028489200832396148683763148742, −7.948597982733608338969925292950, −7.21439356904634772024423946717, −5.03642411618548876173874966014, −4.39917425700914531773270215889, −2.81659243434458661910098368679, 2.38837655985124512019642911157, 3.48549463444657931005356619224, 5.16235074788230816556353484588, 6.68959653974407040274764864586, 8.006030264182891579282044709394, 8.617405781623601843241800653804, 10.51804183926324582074859082667, 11.21928690458408959490371105574, 12.24651945513242530077770155868, 13.34702505321402299987331004263

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