Properties

Label 2-819-91.76-c1-0-4
Degree $2$
Conductor $819$
Sign $-0.486 + 0.873i$
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  + (−0.687 + 2.56i)2-s + (−4.37 − 2.52i)4-s + (1.17 − 1.17i)5-s + (−1.53 + 2.15i)7-s + (5.73 − 5.73i)8-s + (2.21 + 3.83i)10-s + (4.50 + 1.20i)11-s + (−3.59 + 0.264i)13-s + (−4.46 − 5.42i)14-s + (5.71 + 9.90i)16-s + (−3.15 + 5.46i)17-s + (−1.09 − 4.08i)19-s + (−8.13 + 2.17i)20-s + (−6.19 + 10.7i)22-s + (−3.67 + 2.12i)23-s + ⋯
L(s)  = 1  + (−0.486 + 1.81i)2-s + (−2.18 − 1.26i)4-s + (0.526 − 0.526i)5-s + (−0.581 + 0.813i)7-s + (2.02 − 2.02i)8-s + (0.699 + 1.21i)10-s + (1.35 + 0.363i)11-s + (−0.997 + 0.0733i)13-s + (−1.19 − 1.44i)14-s + (1.42 + 2.47i)16-s + (−0.765 + 1.32i)17-s + (−0.250 − 0.936i)19-s + (−1.81 + 0.487i)20-s + (−1.31 + 2.28i)22-s + (−0.766 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.486 + 0.873i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.486 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270227 - 0.459794i\)
\(L(\frac12)\) \(\approx\) \(0.270227 - 0.459794i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 - 2.15i)T \)
13 \( 1 + (3.59 - 0.264i)T \)
good2 \( 1 + (0.687 - 2.56i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.17 + 1.17i)T - 5iT^{2} \)
11 \( 1 + (-4.50 - 1.20i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.15 - 5.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 + 4.08i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.526 - 0.912i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.61 - 5.61i)T - 31iT^{2} \)
37 \( 1 + (-0.572 - 0.153i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.24 + 0.333i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (9.27 + 5.35i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.85 + 2.85i)T + 47iT^{2} \)
53 \( 1 - 0.398T + 53T^{2} \)
59 \( 1 + (-8.26 + 2.21i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.22 - 2.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.76 - 10.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.31 - 1.15i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.935 + 0.935i)T + 73iT^{2} \)
79 \( 1 + 0.927T + 79T^{2} \)
83 \( 1 + (7.79 - 7.79i)T - 83iT^{2} \)
89 \( 1 + (1.28 - 4.78i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.65 + 6.18i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.18631366815394234734079508694, −9.530210382968812039421466321770, −8.887960027461516993889108340303, −8.464197234294004365799871642299, −7.05281859310830606561390091328, −6.66447312152562391800267792381, −5.72521589294590974443488065498, −5.03249035684787269322155109799, −3.94463495497020047541510890270, −1.76511089300785247206053171626, 0.30846511359571370286703506223, 1.82700533829552709194410672586, 2.84777961961617280682064554183, 3.81529348216232269211229281245, 4.61065940150112078290903769698, 6.22765225369844554563924236461, 7.21364434809199228152739458580, 8.331895021167729740205758381958, 9.408082785302373502875052940207, 9.777503028091168083129919540898

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