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Properties

Label 2-819-91.59-c1-0-32
Degree $2$
Conductor $819$
Sign $0.883 - 0.468i$
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  + (0.876 + 0.876i)2-s − 0.463i·4-s + (2.51 + 0.674i)5-s + (2.20 + 1.46i)7-s + (2.15 − 2.15i)8-s + (1.61 + 2.79i)10-s + (−1.36 − 0.365i)11-s + (0.445 − 3.57i)13-s + (0.642 + 3.21i)14-s + 2.85·16-s − 2.82·17-s + (1.61 + 6.04i)19-s + (0.312 − 1.16i)20-s + (−0.875 − 1.51i)22-s − 1.01i·23-s + ⋯
L(s)  = 1  + (0.619 + 0.619i)2-s − 0.231i·4-s + (1.12 + 0.301i)5-s + (0.831 + 0.554i)7-s + (0.763 − 0.763i)8-s + (0.510 + 0.884i)10-s + (−0.411 − 0.110i)11-s + (0.123 − 0.992i)13-s + (0.171 + 0.859i)14-s + 0.714·16-s − 0.685·17-s + (0.371 + 1.38i)19-s + (0.0698 − 0.260i)20-s + (−0.186 − 0.323i)22-s − 0.212i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.883 - 0.468i$
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: \((2,\ 819,\ (\ :1/2),\ 0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74473 + 0.682862i\)
\(L(\frac12)\) \(\approx\) \(2.74473 + 0.682862i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 1.46i)T \)
13 \( 1 + (-0.445 + 3.57i)T \)
good2 \( 1 + (-0.876 - 0.876i)T + 2iT^{2} \)
5 \( 1 + (-2.51 - 0.674i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.36 + 0.365i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + (-1.61 - 6.04i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 1.01iT - 23T^{2} \)
29 \( 1 + (-2.66 + 4.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.73 + 6.46i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.50 - 6.50i)T - 37iT^{2} \)
41 \( 1 + (-2.51 - 9.40i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.850 + 0.490i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.594 - 2.21i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.52 + 4.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.39 - 5.39i)T + 59iT^{2} \)
61 \( 1 + (6.75 + 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.38 - 12.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.97 - 11.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.43 + 2.26i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.78 + 4.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.445 + 0.445i)T - 83iT^{2} \)
89 \( 1 + (0.108 + 0.108i)T + 89iT^{2} \)
97 \( 1 + (2.87 + 0.771i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.18113639115849192385776030207, −9.703700322877694461322524807783, −8.399863482010480297059747549325, −7.69534349156863557185445928324, −6.47277775163343981285170834332, −5.78553549536846783759470089647, −5.31340310923349333701109934295, −4.23091967575260896451243633119, −2.66535319363026050777890617250, −1.50714719236415696414819436262, 1.60476579707986117222352265378, 2.41515668557588195403303551423, 3.77605124773812981231238866802, 4.83620533260601848004921192569, 5.28051923582676185226234371970, 6.76375116922831834424082959315, 7.48511464925921835191179646865, 8.733361846253397721311301898631, 9.214656799824058583136665623536, 10.60072428799341180992896740110

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