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Properties

Label 2-1386-33.2-c1-0-3
Degree $2$
Conductor $1386$
Sign $-0.696 - 0.717i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
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  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−3.41 + 1.10i)5-s + (0.587 − 0.809i)7-s + (0.809 − 0.587i)8-s − 3.58i·10-s + (3.26 − 0.582i)11-s + (2.13 + 0.692i)13-s + (0.587 + 0.809i)14-s + (0.309 + 0.951i)16-s + (0.427 + 1.31i)17-s + (−3.86 − 5.32i)19-s + (3.41 + 1.10i)20-s + (−0.455 + 3.28i)22-s + 7.59i·23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (−1.52 + 0.495i)5-s + (0.222 − 0.305i)7-s + (0.286 − 0.207i)8-s − 1.13i·10-s + (0.984 − 0.175i)11-s + (0.590 + 0.191i)13-s + (0.157 + 0.216i)14-s + (0.0772 + 0.237i)16-s + (0.103 + 0.319i)17-s + (−0.887 − 1.22i)19-s + (0.762 + 0.247i)20-s + (−0.0970 + 0.700i)22-s + 1.58i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8197661273\)
\(L(\frac12)\) \(\approx\) \(0.8197661273\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-3.26 + 0.582i)T \)
good5 \( 1 + (3.41 - 1.10i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-2.13 - 0.692i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.427 - 1.31i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.86 + 5.32i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.59iT - 23T^{2} \)
29 \( 1 + (-0.0255 - 0.0185i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.52 + 4.68i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.90 - 5.01i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.00 - 4.36i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 + (0.0507 + 0.0698i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.10 - 0.683i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.98 - 2.73i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.970 - 0.315i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + (14.6 - 4.76i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.79 - 5.22i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.16 - 1.35i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.14 - 9.69i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 5.35iT - 89T^{2} \)
97 \( 1 + (-0.615 + 1.89i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.699053405062051148304893742066, −8.858406410993776794926074690686, −8.109384393607233124710514539662, −7.51058868129817093753455570494, −6.72045734746006542305388437164, −6.05849994867186482885912756006, −4.59257256814022051409102282831, −4.07010587726958245744736326683, −3.11346554529803456590139430230, −1.17575458432214083379205342966, 0.43448121916464223628855535801, 1.77467631174789932909704506683, 3.25170660642419305486784110815, 4.08259434983101141711817364078, 4.61673339466252574183607188932, 5.93212766089271704098055263148, 7.02396560723668446044260180811, 7.930604976638168833172543005379, 8.650835177442876108847659133735, 8.957077884940981893348722710284

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