↑

Properties

Label 2-819-91.16-c1-0-10
Degree $2$
Conductor $819$
Sign $0.654 - 0.756i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
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  − 1.55·2-s + 0.417·4-s + (−0.595 + 1.03i)5-s + (−2.44 + 1.01i)7-s + 2.46·8-s + (0.926 − 1.60i)10-s + (1.05 − 1.83i)11-s + (2.86 − 2.19i)13-s + (3.79 − 1.58i)14-s − 4.66·16-s + 0.906·17-s + (−3.34 − 5.79i)19-s + (−0.248 + 0.430i)20-s + (−1.64 + 2.84i)22-s − 3.59·23-s + ⋯
L(s)  = 1  − 1.09·2-s + 0.208·4-s + (−0.266 + 0.461i)5-s + (−0.922 + 0.385i)7-s + 0.870·8-s + (0.292 − 0.507i)10-s + (0.319 − 0.552i)11-s + (0.793 − 0.608i)13-s + (1.01 − 0.423i)14-s − 1.16·16-s + 0.219·17-s + (−0.767 − 1.32i)19-s + (−0.0555 + 0.0962i)20-s + (−0.350 + 0.607i)22-s − 0.750·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.654 - 0.756i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.654 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560762 + 0.256442i\)
\(L(\frac12)\) \(\approx\) \(0.560762 + 0.256442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.44 - 1.01i)T \)
13 \( 1 + (-2.86 + 2.19i)T \)
good2 \( 1 + 1.55T + 2T^{2} \)
5 \( 1 + (0.595 - 1.03i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.05 + 1.83i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.906T + 17T^{2} \)
19 \( 1 + (3.34 + 5.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.59T + 23T^{2} \)
29 \( 1 + (-4.25 - 7.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.64 - 4.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
41 \( 1 + (-0.768 - 1.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.71 - 4.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.59 - 2.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.41 + 2.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + (-4.13 - 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.87 + 3.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.26 - 2.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.86 - 4.96i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.03 - 5.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 + (3.10 - 5.37i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.31868809862057861488522800444, −9.372266267213191598694966893748, −8.726726428631312491401210397556, −8.089438238338649794428421119898, −6.93232307485638664686380815541, −6.35729526201917453635043787213, −5.06638612035671248824457711519, −3.72630291882622442005902803108, −2.73557202657388232337604905395, −0.951515938657172648325185370459, 0.61492725103183307822837622676, 2.02138691172068749976143588922, 3.87935818066446225637372566813, 4.41381296632373841035056064698, 6.04475384977785222221563183172, 6.77028799806787954906410329302, 7.929024608875820370595550712883, 8.387050680379619522586512751572, 9.369851633179151988307325149451, 9.965034557380645771543320263692

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