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Properties

Label 2-1386-21.5-c1-0-10
Degree $2$
Conductor $1386$
Sign $0.807 + 0.589i$
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  + (−0.866 − 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: $0.807 + 0.589i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (89, \cdot )$
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

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

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