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Properties

Label 2-819-13.9-c1-0-29
Degree $2$
Conductor $819$
Sign $-0.664 + 0.746i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)2-s + (−0.5 − 0.866i)4-s − 1.73·5-s + (−0.5 − 0.866i)7-s + 1.73·8-s + (−1.49 + 2.59i)10-s + (0.633 − 1.09i)11-s + (3.59 − 0.232i)13-s − 1.73·14-s + (2.49 − 4.33i)16-s + (−3.86 − 6.69i)17-s + (−1 − 1.73i)19-s + (0.866 + 1.50i)20-s + (−1.09 − 1.90i)22-s + (2.36 − 4.09i)23-s + ⋯
L(s)  = 1  + (0.612 − 1.06i)2-s + (−0.250 − 0.433i)4-s − 0.774·5-s + (−0.188 − 0.327i)7-s + 0.612·8-s + (−0.474 + 0.821i)10-s + (0.191 − 0.331i)11-s + (0.997 − 0.0643i)13-s − 0.462·14-s + (0.624 − 1.08i)16-s + (−0.937 − 1.62i)17-s + (−0.229 − 0.397i)19-s + (0.193 + 0.335i)20-s + (−0.234 − 0.405i)22-s + (0.493 − 0.854i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.664 + 0.746i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.664 + 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762260 - 1.69917i\)
\(L(\frac12)\) \(\approx\) \(0.762260 - 1.69917i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.59 + 0.232i)T \)
good2 \( 1 + (-0.866 + 1.5i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 + (-0.633 + 1.09i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.86 + 6.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.36 + 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.19T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.59 - 4.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0980 - 0.169i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 - 9.92T + 53T^{2} \)
59 \( 1 + (-3.63 - 6.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.40 - 4.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.09 + 5.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.19T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 2.19T + 83T^{2} \)
89 \( 1 + (-6.46 + 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.19 + 5.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.24330830276711452525573613782, −9.195158521482111146577365787089, −8.292155643609605566056909146328, −7.30891599084988899951216624108, −6.49029079214432068725010745250, −5.03009239526639397128313806933, −4.22843279020263432811690415438, −3.41776399418811759011358789549, −2.45773517442930590152149131707, −0.795358460552956475064080498450, 1.74197926939890262643808917422, 3.65084687526249175622769557865, 4.24279220095949821797261467522, 5.38688298048962016830168346771, 6.31367153352645821848031143752, 6.80515799414133270797056694387, 8.085578409298166374668235781995, 8.313508902216873771799654165412, 9.612507995338872836247114248582, 10.66269085391862428470021320100

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