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Properties

Label 2-819-91.59-c1-0-26
Degree $2$
Conductor $819$
Sign $0.844 - 0.535i$
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  + (1.42 + 1.42i)2-s + 2.06i·4-s + (−3.16 − 0.849i)5-s + (−0.111 − 2.64i)7-s + (−0.0872 + 0.0872i)8-s + (−3.30 − 5.72i)10-s + (5.74 + 1.53i)11-s + (2.81 + 2.25i)13-s + (3.60 − 3.92i)14-s + 3.87·16-s + 0.628·17-s + (−0.191 − 0.712i)19-s + (1.75 − 6.53i)20-s + (5.99 + 10.3i)22-s − 4.54i·23-s + ⋯
L(s)  = 1  + (1.00 + 1.00i)2-s + 1.03i·4-s + (−1.41 − 0.379i)5-s + (−0.0421 − 0.999i)7-s + (−0.0308 + 0.0308i)8-s + (−1.04 − 1.81i)10-s + (1.73 + 0.463i)11-s + (0.779 + 0.626i)13-s + (0.964 − 1.04i)14-s + 0.968·16-s + 0.152·17-s + (−0.0438 − 0.163i)19-s + (0.391 − 1.46i)20-s + (1.27 + 2.21i)22-s − 0.948i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27349 + 0.659647i\)
\(L(\frac12)\) \(\approx\) \(2.27349 + 0.659647i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.111 + 2.64i)T \)
13 \( 1 + (-2.81 - 2.25i)T \)
good2 \( 1 + (-1.42 - 1.42i)T + 2iT^{2} \)
5 \( 1 + (3.16 + 0.849i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-5.74 - 1.53i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.628T + 17T^{2} \)
19 \( 1 + (0.191 + 0.712i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.54iT - 23T^{2} \)
29 \( 1 + (-1.33 + 2.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.285 + 1.06i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.79 + 1.79i)T - 37iT^{2} \)
41 \( 1 + (-0.746 - 2.78i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.49 + 2.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.90 + 7.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.89 - 6.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.673 + 0.673i)T + 59iT^{2} \)
61 \( 1 + (0.943 + 0.544i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.39 - 5.21i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.590 - 2.20i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.94 - 1.59i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.08 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.59 - 3.59i)T - 83iT^{2} \)
89 \( 1 + (-2.88 - 2.88i)T + 89iT^{2} \)
97 \( 1 + (-8.41 - 2.25i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.42509232235091768494532151806, −9.216451694523055374907990325565, −8.309453785057686311783055824889, −7.43143240455568506615054985467, −6.83328144980629629274994794145, −6.11936543754812580325002130056, −4.57658085147876462720833914313, −4.18557497084401436678164582790, −3.61082127559051310908586909281, −1.08122461081376860081287304397, 1.38917106709600750063732996120, 3.02191066323641264556779140007, 3.57961453652407424675512295171, 4.34558291235941256136676372217, 5.56359457434245012991086570176, 6.42707151836524647752392563054, 7.68222449135793993070059046661, 8.490640061520079767364822288277, 9.369919586955789221427925078806, 10.64352057550311512163305048829

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