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Properties

Label 2-819-91.89-c1-0-40
Degree $2$
Conductor $819$
Sign $-0.610 + 0.792i$
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  + (0.270 + 0.270i)2-s − 1.85i·4-s + (−0.959 − 3.58i)5-s + (1.30 − 2.30i)7-s + (1.04 − 1.04i)8-s + (0.709 − 1.22i)10-s + (−0.0226 − 0.0846i)11-s + (1.63 + 3.21i)13-s + (0.975 − 0.270i)14-s − 3.14·16-s + 5.89·17-s + (−3.58 − 0.960i)19-s + (−6.63 + 1.77i)20-s + (0.0167 − 0.0290i)22-s − 0.446i·23-s + ⋯
L(s)  = 1  + (0.191 + 0.191i)2-s − 0.926i·4-s + (−0.429 − 1.60i)5-s + (0.492 − 0.870i)7-s + (0.368 − 0.368i)8-s + (0.224 − 0.388i)10-s + (−0.00683 − 0.0255i)11-s + (0.453 + 0.891i)13-s + (0.260 − 0.0722i)14-s − 0.785·16-s + 1.42·17-s + (−0.822 − 0.220i)19-s + (−1.48 + 0.397i)20-s + (0.00357 − 0.00619i)22-s − 0.0930i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.610 + 0.792i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.698016 - 1.41833i\)
\(L(\frac12)\) \(\approx\) \(0.698016 - 1.41833i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 2.30i)T \)
13 \( 1 + (-1.63 - 3.21i)T \)
good2 \( 1 + (-0.270 - 0.270i)T + 2iT^{2} \)
5 \( 1 + (0.959 + 3.58i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0226 + 0.0846i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 + (3.58 + 0.960i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 0.446iT - 23T^{2} \)
29 \( 1 + (0.706 + 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.94 + 0.520i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.87 + 1.87i)T - 37iT^{2} \)
41 \( 1 + (3.00 + 0.804i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.64 - 4.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.84 - 2.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.28 - 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.05 + 5.05i)T + 59iT^{2} \)
61 \( 1 + (-0.110 + 0.0638i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.61 + 2.57i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-9.83 + 2.63i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.37 - 8.84i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.75 + 3.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.17 + 2.17i)T - 83iT^{2} \)
89 \( 1 + (1.19 + 1.19i)T + 89iT^{2} \)
97 \( 1 + (0.452 + 1.68i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.816652912067718015979609615493, −9.133382611576421248205848841730, −8.233493834587080697196246880426, −7.44530746010379061386973757922, −6.30005546184464755200942999966, −5.33964708517177271061584621407, −4.55764234501995742280153312607, −3.94041846943656933695005454250, −1.66433117737984582910370301789, −0.78087800567129354785089878561, 2.22375435933633960163919880929, 3.16465861430244262945547086586, 3.77571745182878917560811531555, 5.22877110497182214170080135257, 6.24376818061040725764757559619, 7.24919746013958632761524394863, 7.963861838587685708306842524520, 8.542097499233560262872225592500, 9.882474589241238228023767211838, 10.77315056188526580629622214705

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