Properties

Degree $2$
Conductor $1386$
Sign $0.807 - 0.589i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.0340 + 0.0590i)5-s + (−2.19 + 1.48i)7-s + 0.999i·8-s + (−0.0590 − 0.0340i)10-s + (−0.866 − 0.5i)11-s − 2.44i·13-s + (1.15 − 2.38i)14-s + (−0.5 − 0.866i)16-s + (−0.817 + 1.41i)17-s + (6.28 − 3.62i)19-s + 0.0681·20-s + 0.999·22-s + (−0.405 + 0.233i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0152 + 0.0263i)5-s + (−0.827 + 0.560i)7-s + 0.353i·8-s + (−0.0186 − 0.0107i)10-s + (−0.261 − 0.150i)11-s − 0.679i·13-s + (0.308 − 0.636i)14-s + (−0.125 − 0.216i)16-s + (−0.198 + 0.343i)17-s + (1.44 − 0.832i)19-s + 0.0152·20-s + 0.213·22-s + (−0.0844 + 0.0487i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.807 - 0.589i$
Motivic weight: \(1\)
Character: $\chi_{1386} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.807 - 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.049283179\)
\(L(\frac12)\) \(\approx\) \(1.049283179\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (2.19 - 1.48i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-0.0340 - 0.0590i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (0.817 - 1.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.28 + 3.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.405 - 0.233i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.23iT - 29T^{2} \)
31 \( 1 + (0.201 + 0.116i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.74 - 4.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.38T + 41T^{2} \)
43 \( 1 - 9.64T + 43T^{2} \)
47 \( 1 + (-4.31 - 7.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.34 - 4.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.439 - 0.760i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.63 - 4.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.52 + 2.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 + (-6.76 - 3.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.76 - 9.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + (-5.33 - 9.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.99iT - 97T^{2} \)
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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

−9.430888034658628789965556044720, −8.990718614279377723079638030287, −8.064888609109936508059847412130, −7.27363042339905237491006976137, −6.43695799447843433256731935242, −5.67224830968862221825978454976, −4.83766164608747653543670574780, −3.31889641775730839643376172842, −2.54249615772210774631468221955, −0.867751934221223898710504501988, 0.75538822869407592697024860327, 2.17521287269402542179947594059, 3.32386733151507176540267794291, 4.10966042269207682219255683872, 5.37281168744074261285009442664, 6.38576189370233411156094302232, 7.27617458955811080473404533169, 7.76416539064474246709114293438, 8.957519951047449062750325330263, 9.535388700749631801783190881723

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