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Properties

Label 2-819-117.103-c1-0-0
Degree $2$
Conductor $819$
Sign $-0.987 + 0.159i$
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  + (0.654 − 0.377i)2-s + (−1.67 + 0.442i)3-s + (−0.714 + 1.23i)4-s + (−0.788 − 0.455i)5-s + (−0.928 + 0.921i)6-s + (0.866 − 0.5i)7-s + 2.59i·8-s + (2.60 − 1.48i)9-s − 0.687·10-s + (−0.550 + 0.317i)11-s + (0.649 − 2.38i)12-s + (−0.0262 + 3.60i)13-s + (0.377 − 0.654i)14-s + (1.52 + 0.413i)15-s + (−0.451 − 0.782i)16-s − 5.38·17-s + ⋯
L(s)  = 1  + (0.462 − 0.267i)2-s + (−0.966 + 0.255i)3-s + (−0.357 + 0.619i)4-s + (−0.352 − 0.203i)5-s + (−0.378 + 0.376i)6-s + (0.327 − 0.188i)7-s + 0.915i·8-s + (0.869 − 0.493i)9-s − 0.217·10-s + (−0.165 + 0.0958i)11-s + (0.187 − 0.689i)12-s + (−0.00728 + 0.999i)13-s + (0.100 − 0.174i)14-s + (0.392 + 0.106i)15-s + (−0.112 − 0.195i)16-s − 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.987 + 0.159i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.987 + 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00971348 - 0.121049i\)
\(L(\frac12)\) \(\approx\) \(0.00971348 - 0.121049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.442i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.0262 - 3.60i)T \)
good2 \( 1 + (-0.654 + 0.377i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.788 + 0.455i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.550 - 0.317i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 5.38T + 17T^{2} \)
19 \( 1 + 0.0581iT - 19T^{2} \)
23 \( 1 + (-1.10 + 1.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.74 + 3.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.59 + 1.50i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.57iT - 37T^{2} \)
41 \( 1 + (5.78 + 3.34i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.31 + 5.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (11.5 - 6.65i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.995T + 53T^{2} \)
59 \( 1 + (8.00 + 4.62i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.30 + 2.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 2.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.95iT - 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 + (-1.05 - 1.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.65 + 2.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.50iT - 89T^{2} \)
97 \( 1 + (-10.6 + 6.17i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.05737659502512501496410377739, −9.941326411382001814765758711385, −8.991331050743931321593363336064, −8.170950455468578301665284461268, −7.12955920830316683799953779661, −6.27664128931078835779630144765, −4.96297634151095139129279164498, −4.49123757692490282250429105532, −3.68278101450813463223284318104, −2.04802836819044786145615780310, 0.05797507797693691891871799339, 1.64741772941861096269619085481, 3.50961661844326746178141178639, 4.73019922068992384571612112135, 5.28877440293864289798229375219, 6.15705978949261101433020984853, 6.96617249264357181901446668399, 7.85320116350627135293579293077, 9.006233262561055124706285097722, 9.980328934797278954818975959548

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