Properties

Label 2-819-91.20-c1-0-21
Degree $2$
Conductor $819$
Sign $0.755 - 0.655i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 − 0.299i)2-s + (−0.576 + 0.332i)4-s + (−0.549 + 0.549i)5-s + (1.61 − 2.09i)7-s + (−2.17 + 2.17i)8-s + (−0.448 + 0.777i)10-s + (0.824 + 3.07i)11-s + (2.63 + 2.45i)13-s + (1.17 − 2.82i)14-s + (−1.11 + 1.92i)16-s + (−1.74 − 3.03i)17-s + (6.06 + 1.62i)19-s + (0.133 − 0.499i)20-s + (1.83 + 3.18i)22-s + (4.89 + 2.82i)23-s + ⋯
L(s)  = 1  + (0.789 − 0.211i)2-s + (−0.288 + 0.166i)4-s + (−0.245 + 0.245i)5-s + (0.609 − 0.792i)7-s + (−0.769 + 0.769i)8-s + (−0.141 + 0.245i)10-s + (0.248 + 0.927i)11-s + (0.731 + 0.682i)13-s + (0.313 − 0.754i)14-s + (−0.278 + 0.482i)16-s + (−0.424 − 0.735i)17-s + (1.39 + 0.372i)19-s + (0.0299 − 0.111i)20-s + (0.392 + 0.679i)22-s + (1.01 + 0.588i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90848 + 0.712316i\)
\(L(\frac12)\) \(\approx\) \(1.90848 + 0.712316i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 2.09i)T \)
13 \( 1 + (-2.63 - 2.45i)T \)
good2 \( 1 + (-1.11 + 0.299i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.549 - 0.549i)T - 5iT^{2} \)
11 \( 1 + (-0.824 - 3.07i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.74 + 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.06 - 1.62i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.54 - 7.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.888 - 0.888i)T - 31iT^{2} \)
37 \( 1 + (-0.151 - 0.564i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.704 + 2.63i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.60 + 3.81i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.267 + 0.267i)T + 47iT^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (0.635 - 2.37i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.70 - 3.86i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.90 - 1.85i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.51 + 9.37i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.71 + 7.71i)T + 73iT^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + (10.3 - 10.3i)T - 83iT^{2} \)
89 \( 1 + (-7.02 + 1.88i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.704 + 0.188i)T + (84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.56010161715600965248378214755, −9.315133509424468080489754106075, −8.841339974565670187751792379679, −7.35517799265530442230383985704, −7.21067469403562810005752215623, −5.59758011545819374998283969745, −4.82293647785384962690949003542, −3.95979223165864041068954839181, −3.17206302073787479345761934265, −1.53212125931785883708034946123, 0.902964110116331955454237140807, 2.80613644257978602720588358319, 3.88245103048895977131291327539, 4.80781577158760012970742012708, 5.74994841963110473278858137629, 6.17707501211548536794547912965, 7.61449440136241354732503385572, 8.611152904261241260174152878988, 9.018295791821864886917170755999, 10.14878634224077200032073631349

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