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Properties

Label 2-819-91.4-c1-0-22
Degree $2$
Conductor $819$
Sign $0.946 - 0.323i$
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  + 2.50i·2-s − 4.27·4-s + (−2.61 + 1.50i)5-s + (−2.46 − 0.967i)7-s − 5.71i·8-s + (−3.77 − 6.54i)10-s + (1.34 − 0.775i)11-s + (−0.822 − 3.51i)13-s + (2.42 − 6.17i)14-s + 5.75·16-s + 1.51·17-s + (7.11 + 4.10i)19-s + (11.1 − 6.45i)20-s + (1.94 + 3.36i)22-s − 3.64·23-s + ⋯
L(s)  = 1  + 1.77i·2-s − 2.13·4-s + (−1.16 + 0.674i)5-s + (−0.930 − 0.365i)7-s − 2.01i·8-s + (−1.19 − 2.07i)10-s + (0.405 − 0.233i)11-s + (−0.228 − 0.973i)13-s + (0.647 − 1.64i)14-s + 1.43·16-s + 0.367·17-s + (1.63 + 0.942i)19-s + (2.49 − 1.44i)20-s + (0.414 + 0.717i)22-s − 0.760·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.946 - 0.323i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.946 - 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418768 + 0.0695001i\)
\(L(\frac12)\) \(\approx\) \(0.418768 + 0.0695001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.967i)T \)
13 \( 1 + (0.822 + 3.51i)T \)
good2 \( 1 - 2.50iT - 2T^{2} \)
5 \( 1 + (2.61 - 1.50i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.34 + 0.775i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.51T + 17T^{2} \)
19 \( 1 + (-7.11 - 4.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.64T + 23T^{2} \)
29 \( 1 + (2.75 - 4.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.47 + 2.58i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.20iT - 37T^{2} \)
41 \( 1 + (2.85 + 1.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.97 + 3.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.00 + 4.62i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.90 - 6.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + (-5.23 + 9.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.52 + 1.45i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.11 + 0.642i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.27 + 1.31i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.28 + 5.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.49iT - 83T^{2} \)
89 \( 1 + 9.76iT - 89T^{2} \)
97 \( 1 + (-10.8 + 6.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.989587348097973507813902937568, −9.226062456692819918126504508359, −8.138667421389930232875618412434, −7.44021378006414366737619694647, −7.13718810484305667001269455556, −5.99565180462429846185183886916, −5.34875985027204710634862130532, −3.83390384589151719046728655303, −3.43276353652782672232312551691, −0.25006459171500268568845712239, 1.16850176445562178215597553059, 2.65796562393642820536679577552, 3.64245099321674240712744927634, 4.31496147703642156973860847909, 5.30149816612043068217045996011, 6.83500223232133416432675287044, 7.938116279247647723072227729467, 8.998130409229404965175923428875, 9.441736478725475760603927772373, 10.15552836050450646734933884191

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