↑

Properties

Label 2-1386-7.4-c1-0-16
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

Related objects

Downloads

Learn more

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: $\chi_{1386} (991, \cdot )$
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 \( 1 - 7T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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