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Properties

Label 2-1386-7.2-c1-0-14
Degree $2$
Conductor $1386$
Sign $0.605 - 0.795i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (2 + 1.73i)7-s + 0.999·8-s + (−0.5 − 0.866i)11-s + 4·13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (1.5 − 2.59i)19-s + 0.999·22-s + (−0.5 + 0.866i)23-s + (2.5 + 4.33i)25-s + (−2 + 3.46i)26-s + (0.5 − 2.59i)28-s + 29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.755 + 0.654i)7-s + 0.353·8-s + (−0.150 − 0.261i)11-s + 1.10·13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.121 − 0.210i)17-s + (0.344 − 0.596i)19-s + 0.213·22-s + (−0.104 + 0.180i)23-s + (0.5 + 0.866i)25-s + (−0.392 + 0.679i)26-s + (0.0944 − 0.490i)28-s + 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.551934295\)
\(L(\frac12)\) \(\approx\) \(1.551934295\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + (6 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.236724046381275993647095006301, −9.002338443464102858704753666565, −8.029681256908266097291219361674, −7.44181061763764898260085349480, −6.35513047013373148306125061944, −5.63451907964691722057356383326, −4.88413782263441815320906466744, −3.74910637044342925611170918311, −2.40532938657690266230165235585, −1.07225671076757514254252704793, 0.961111099051834868054928552456, 2.00178093046359585047280429132, 3.34365604398416506472396542466, 4.19118632286208238114467399023, 5.06435612130953063479283479425, 6.23161248907840701119520015073, 7.18304020247927490510410013078, 8.104830595294451676100375430107, 8.545816587868687200787233232187, 9.565120466471218618658680802780

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