↑

Properties

Label 2-819-13.4-c1-0-1
Degree $2$
Conductor $819$
Sign $-0.665 - 0.746i$
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  + (−1.82 − 1.05i)2-s + (1.22 + 2.12i)4-s + 3.60i·5-s + (−0.866 + 0.5i)7-s − 0.948i·8-s + (3.79 − 6.57i)10-s + (−0.767 − 0.443i)11-s + (−1.17 + 3.40i)13-s + 2.10·14-s + (1.44 − 2.51i)16-s + (2.48 + 4.29i)17-s + (2.06 − 1.18i)19-s + (−7.64 + 4.41i)20-s + (0.934 + 1.61i)22-s + (1.92 − 3.34i)23-s + ⋯
L(s)  = 1  + (−1.29 − 0.745i)2-s + (0.612 + 1.06i)4-s + 1.61i·5-s + (−0.327 + 0.188i)7-s − 0.335i·8-s + (1.20 − 2.08i)10-s + (−0.231 − 0.133i)11-s + (−0.325 + 0.945i)13-s + 0.563·14-s + (0.362 − 0.627i)16-s + (0.601 + 1.04i)17-s + (0.472 − 0.272i)19-s + (−1.70 + 0.986i)20-s + (0.199 + 0.345i)22-s + (0.402 − 0.696i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.665 - 0.746i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160650 + 0.358641i\)
\(L(\frac12)\) \(\approx\) \(0.160650 + 0.358641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (1.17 - 3.40i)T \)
good2 \( 1 + (1.82 + 1.05i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 3.60iT - 5T^{2} \)
11 \( 1 + (0.767 + 0.443i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.48 - 4.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.06 + 1.18i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.92 + 3.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.640 + 1.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.46iT - 31T^{2} \)
37 \( 1 + (8.34 + 4.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.4 + 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.82 + 3.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.98iT - 47T^{2} \)
53 \( 1 + 4.92T + 53T^{2} \)
59 \( 1 + (6.34 - 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.769 - 1.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.29 - 4.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.58 + 3.22i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.14iT - 73T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 - 4.76iT - 83T^{2} \)
89 \( 1 + (3.13 + 1.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.401 - 0.231i)T + (48.5 - 84.0i)T^{2} \)
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

−10.51627012203119988504787257720, −9.917275461971044364092460871463, −9.000532123725969600111138135983, −8.218025527423759919744072860684, −7.13565768797154903127172302899, −6.67121364813759219558672184331, −5.35663727661839993834477179310, −3.60119986585884416472557908841, −2.80370026950832172282217443835, −1.78140263584117322120609364631, 0.31272700353628214298337508069, 1.40046873858469506287021430801, 3.37884337282545561139098205089, 4.90957079127457910648110818597, 5.51728011235850988991442722226, 6.71454687279191668641073709512, 7.78000374900939385880275607372, 8.093326688132704015603973451100, 9.096282952504400784100144261386, 9.676396577289669826881771970595

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