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Properties

Label 2-1386-33.8-c1-0-2
Degree $2$
Conductor $1386$
Sign $-0.578 - 0.815i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−1.04 + 1.43i)5-s + (−0.951 − 0.309i)7-s + (0.309 + 0.951i)8-s − 1.77i·10-s + (−0.808 − 3.21i)11-s + (−2.87 − 3.95i)13-s + (0.951 − 0.309i)14-s + (−0.809 − 0.587i)16-s + (3.29 + 2.39i)17-s + (6.36 − 2.06i)19-s + (1.04 + 1.43i)20-s + (2.54 + 2.12i)22-s + 8.52i·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.466 + 0.641i)5-s + (−0.359 − 0.116i)7-s + (0.109 + 0.336i)8-s − 0.561i·10-s + (−0.243 − 0.969i)11-s + (−0.796 − 1.09i)13-s + (0.254 − 0.0825i)14-s + (−0.202 − 0.146i)16-s + (0.798 + 0.580i)17-s + (1.46 − 0.474i)19-s + (0.233 + 0.320i)20-s + (0.542 + 0.453i)22-s + 1.77i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.578 - 0.815i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6675382484\)
\(L(\frac12)\) \(\approx\) \(0.6675382484\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (0.808 + 3.21i)T \)
good5 \( 1 + (1.04 - 1.43i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.87 + 3.95i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.29 - 2.39i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.36 + 2.06i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.52iT - 23T^{2} \)
29 \( 1 + (1.42 - 4.38i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.66 - 4.11i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.200 + 0.617i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.28 - 10.1i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.80iT - 43T^{2} \)
47 \( 1 + (1.05 - 0.344i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.49 + 7.55i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.65 + 1.18i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.09 - 7.01i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.30T + 67T^{2} \)
71 \( 1 + (0.602 - 0.829i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.69 + 1.20i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.18 - 12.6i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.47 + 1.79i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.7iT - 89T^{2} \)
97 \( 1 + (-0.363 + 0.264i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.741660472263690997222139432724, −9.148083778291433717576021671769, −7.923638074897327150427446445773, −7.60458926766387045342894281665, −6.84104361559131662305207348348, −5.64398148565916924025873233547, −5.25015403683486460287145650530, −3.41645726614421900784488921845, −3.08843199610959076349667090523, −1.19808934735798465506426036866, 0.36675621717364068283179069244, 1.87969934579015172149908178614, 2.92987270478042728378885495296, 4.20062404528835276196275446904, 4.83400806952952141475307468660, 6.03966042111624470045903060012, 7.28050308521556917048912893136, 7.58208112348201741729369106922, 8.635033888483751025702468003080, 9.511277570783984860998484129255

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