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Properties

Label 2-819-91.23-c1-0-39
Degree $2$
Conductor $819$
Sign $-0.798 - 0.601i$
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  − 1.70i·2-s − 0.922·4-s + (−1.84 − 1.06i)5-s + (1.43 + 2.22i)7-s − 1.84i·8-s + (−1.81 + 3.15i)10-s + (−2.01 − 1.16i)11-s + (−2.64 − 2.44i)13-s + (3.79 − 2.45i)14-s − 4.99·16-s − 0.207·17-s + (2.00 − 1.16i)19-s + (1.70 + 0.981i)20-s + (−1.98 + 3.44i)22-s − 5.92·23-s + ⋯
L(s)  = 1  − 1.20i·2-s − 0.461·4-s + (−0.824 − 0.475i)5-s + (0.543 + 0.839i)7-s − 0.651i·8-s + (−0.575 + 0.996i)10-s + (−0.607 − 0.350i)11-s + (−0.733 − 0.679i)13-s + (1.01 − 0.656i)14-s − 1.24·16-s − 0.0504·17-s + (0.460 − 0.266i)19-s + (0.380 + 0.219i)20-s + (−0.423 + 0.734i)22-s − 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.798 - 0.601i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.798 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235223 + 0.703303i\)
\(L(\frac12)\) \(\approx\) \(0.235223 + 0.703303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 - 2.22i)T \)
13 \( 1 + (2.64 + 2.44i)T \)
good2 \( 1 + 1.70iT - 2T^{2} \)
5 \( 1 + (1.84 + 1.06i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.01 + 1.16i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.207T + 17T^{2} \)
19 \( 1 + (-2.00 + 1.16i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.92T + 23T^{2} \)
29 \( 1 + (2.10 + 3.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.50 - 2.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.74iT - 37T^{2} \)
41 \( 1 + (2.48 - 1.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.38 - 5.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.6 - 6.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.04 + 3.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.29iT - 59T^{2} \)
61 \( 1 + (0.470 + 0.815i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.52 + 2.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.10 - 0.638i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.60 + 2.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.77 + 9.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.31iT - 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + (-3.07 - 1.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.862084891012662907564992601746, −9.064148059435233678439530695826, −8.065339327989144989176368319551, −7.49988986912040852563659700536, −5.98735185612321543748482962198, −5.00494381574476137180299444297, −4.03339416007483447765193676122, −2.93730664784446241923210580321, −1.99159946904634287335304439713, −0.34204379207149934562673021649, 2.09189955522879662694579335614, 3.67982405457709444762221316542, 4.66197187637091432401841717585, 5.52624603390071428369704946838, 6.75115722495663362813362268519, 7.38988250632586825001588424957, 7.76661759344336950165987398683, 8.675660819958428927910180818717, 9.902086218044447156422548566194, 10.72263986652619652507466397866

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