Properties

Label 2-819-91.33-c1-0-36
Degree $2$
Conductor $819$
Sign $0.497 + 0.867i$
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  + (0.0881 − 0.328i)2-s + (1.63 + 0.942i)4-s + (−0.552 − 2.06i)5-s + (2.31 − 1.28i)7-s + (0.935 − 0.935i)8-s − 0.727·10-s + (0.152 − 0.152i)11-s + (−3.58 + 0.408i)13-s + (−0.219 − 0.873i)14-s + (1.65 + 2.87i)16-s + (2.64 − 4.58i)17-s + (4.54 − 4.54i)19-s + (1.04 − 3.88i)20-s + (−0.0367 − 0.0636i)22-s + (−5.58 + 3.22i)23-s + ⋯
L(s)  = 1  + (0.0623 − 0.232i)2-s + (0.815 + 0.471i)4-s + (−0.247 − 0.922i)5-s + (0.873 − 0.486i)7-s + (0.330 − 0.330i)8-s − 0.230·10-s + (0.0460 − 0.0460i)11-s + (−0.993 + 0.113i)13-s + (−0.0586 − 0.233i)14-s + (0.414 + 0.718i)16-s + (0.641 − 1.11i)17-s + (1.04 − 1.04i)19-s + (0.232 − 0.869i)20-s + (−0.00783 − 0.0135i)22-s + (−1.16 + 0.672i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.497 + 0.867i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.497 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74526 - 1.01103i\)
\(L(\frac12)\) \(\approx\) \(1.74526 - 1.01103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.31 + 1.28i)T \)
13 \( 1 + (3.58 - 0.408i)T \)
good2 \( 1 + (-0.0881 + 0.328i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.552 + 2.06i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.152 + 0.152i)T - 11iT^{2} \)
17 \( 1 + (-2.64 + 4.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.54 + 4.54i)T - 19iT^{2} \)
23 \( 1 + (5.58 - 3.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.21 - 2.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 + 0.681i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.73 + 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.28 - 8.53i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.95 - 2.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-12.3 + 3.31i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.97 + 3.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.8 + 3.16i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 2.81iT - 61T^{2} \)
67 \( 1 + (-1.19 - 1.19i)T + 67iT^{2} \)
71 \( 1 + (0.653 - 2.44i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.00884 + 0.0329i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.28 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.72 - 7.72i)T - 83iT^{2} \)
89 \( 1 + (2.16 - 8.08i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.90 + 2.11i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.10271383634305154219529919801, −9.307096247016275178549418341178, −8.208033705998716598784274826649, −7.52730611093050169440636598620, −6.96634184323481218915460670255, −5.39906190741791772509901579585, −4.73180700969194627613388725614, −3.61526538104799485361127463568, −2.36949776188507655817377392633, −1.05345412683154665228663798608, 1.70405702830551151408916130482, 2.67154106402759313504726982145, 3.95121323744893520963127516725, 5.39694749847102829259302357814, 5.88339547277481069994444823213, 7.07425284778860273930682294988, 7.61600407485164084607030076382, 8.428477710911736095967932105745, 9.835593766331440935761624759007, 10.44314620816606559628298910055

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