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Properties

Label 2-1386-21.17-c1-0-1
Degree $2$
Conductor $1386$
Sign $-0.589 - 0.807i$
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 + (−1.25 − 2.18i)5-s + (1.48 + 2.19i)7-s + 0.999i·8-s + (2.18 + 1.25i)10-s + (−0.866 − 0.5i)11-s − 2.44i·13-s + (−2.38 − 1.15i)14-s + (−0.5 − 0.866i)16-s + (−2.64 + 4.58i)17-s + (−4.52 + 2.61i)19-s − 2.51·20-s + 0.999·22-s + (−4.28 + 2.47i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.562 − 0.975i)5-s + (0.560 + 0.827i)7-s + 0.353i·8-s + (0.689 + 0.398i)10-s + (−0.261 − 0.150i)11-s − 0.679i·13-s + (−0.636 − 0.308i)14-s + (−0.125 − 0.216i)16-s + (−0.641 + 1.11i)17-s + (−1.03 + 0.599i)19-s − 0.562·20-s + 0.213·22-s + (−0.893 + 0.515i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.589 - 0.807i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.589 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5204600965\)
\(L(\frac12)\) \(\approx\) \(0.5204600965\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 2.19i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (1.25 + 2.18i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (2.64 - 4.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.52 - 2.61i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.28 - 2.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.05iT - 29T^{2} \)
31 \( 1 + (-7.44 - 4.29i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.91 - 10.2i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + (0.637 + 1.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.414 + 0.239i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.02 - 5.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.7 + 6.20i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.11 - 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.7iT - 71T^{2} \)
73 \( 1 + (8.52 + 4.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.52 - 6.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.35T + 83T^{2} \)
89 \( 1 + (-7.92 - 13.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−9.849939709525965533820859799709, −8.626594328474351772769549462392, −8.259615980495998386856761575712, −7.992109916460733034620601578711, −6.48433715021587605955343110561, −5.84064399146085167108611601540, −4.87649260633879586537777957524, −4.08289865242954235544859360174, −2.51443991503734483462366951464, −1.33410934104592586783201789441, 0.26687225542029372545484767147, 1.97799981331759210486498864150, 2.95111056490462279502992097660, 4.08700387216366460446715263370, 4.76118258895021516631708781688, 6.41667790570074026500542712793, 7.02161188953433779532948792189, 7.64046419711483072567722179322, 8.469105410722133560786735207760, 9.337440768678084479617600899504

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