Properties

Degree $2$
Conductor $1386$
Sign $-0.781 - 0.624i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−3.19 − 1.84i)5-s + (−1.52 − 2.15i)7-s + 0.999·8-s + (3.19 − 1.84i)10-s + (0.412 + 3.29i)11-s − 5.60i·13-s + (2.63 − 0.245i)14-s + (−0.5 + 0.866i)16-s + (−1.39 − 2.41i)17-s + (−2.14 − 1.23i)19-s + 3.69i·20-s + (−3.05 − 1.28i)22-s + (3.26 + 1.88i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.42 − 0.825i)5-s + (−0.578 − 0.815i)7-s + 0.353·8-s + (1.01 − 0.583i)10-s + (0.124 + 0.992i)11-s − 1.55i·13-s + (0.704 − 0.0656i)14-s + (−0.125 + 0.216i)16-s + (−0.338 − 0.585i)17-s + (−0.492 − 0.284i)19-s + 0.825i·20-s + (−0.651 − 0.274i)22-s + (0.680 + 0.392i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.781 - 0.624i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.781 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08157599825\)
\(L(\frac12)\) \(\approx\) \(0.08157599825\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (1.52 + 2.15i)T \)
11 \( 1 + (-0.412 - 3.29i)T \)
good5 \( 1 + (3.19 + 1.84i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 5.60iT - 13T^{2} \)
17 \( 1 + (1.39 + 2.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.14 + 1.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.26 - 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.51T + 29T^{2} \)
31 \( 1 + (1.70 + 2.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.85 + 4.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 7.72iT - 43T^{2} \)
47 \( 1 + (-8.98 - 5.18i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.27 - 5.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.6 - 6.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.13 + 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.66 - 9.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.15iT - 71T^{2} \)
73 \( 1 + (4.53 - 2.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.250 + 0.144i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (-13.7 - 7.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.58T + 97T^{2} \)
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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

−9.660432258912941220521732600590, −9.039834159974058828480512170234, −8.046538191958766459081008666538, −7.52721433077990459654470816531, −7.00462174391409477390485605832, −5.77851822523925265770577356282, −4.70976498246521428088114175122, −4.17759917777756416132369127584, −3.04218645515034403392386233281, −1.00238124942084706913503764984, 0.04679359025663189343132581812, 2.01276711321343140177604759677, 3.21129517958780044378559664040, 3.71785085162487272154433625651, 4.74761897911049648295298140919, 6.26739930746099925562799570081, 6.78997254981652311370380163180, 7.79031292669865340305497035032, 8.715592750495988044007651129167, 8.999984907675847781383349339041

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