Properties

Label 2-1638-13.10-c1-0-24
Degree $2$
Conductor $1638$
Sign $0.0634 + 0.997i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 3.62i·5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (1.81 + 3.13i)10-s + (1.74 − 1.00i)11-s + (3.59 − 0.275i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (3.79 − 6.56i)17-s + (2.40 + 1.38i)19-s + (−3.13 − 1.81i)20-s + (−1.00 + 1.74i)22-s + (3.95 + 6.85i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 1.62i·5-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (0.572 + 0.992i)10-s + (0.526 − 0.303i)11-s + (0.997 − 0.0762i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (0.919 − 1.59i)17-s + (0.551 + 0.318i)19-s + (−0.701 − 0.405i)20-s + (−0.214 + 0.372i)22-s + (0.824 + 1.42i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0634 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0634 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.0634 + 0.997i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.0634 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.305445402\)
\(L(\frac12)\) \(\approx\) \(1.305445402\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-3.59 + 0.275i)T \)
good5 \( 1 + 3.62iT - 5T^{2} \)
11 \( 1 + (-1.74 + 1.00i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.79 + 6.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.40 - 1.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.95 - 6.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.83 + 4.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.83iT - 31T^{2} \)
37 \( 1 + (1.48 - 0.859i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.0 + 5.81i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.63 + 4.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.63iT - 47T^{2} \)
53 \( 1 + 0.0731T + 53T^{2} \)
59 \( 1 + (1.15 + 0.667i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.187 + 0.325i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.3 - 7.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.0 + 6.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.80iT - 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 1.97iT - 83T^{2} \)
89 \( 1 + (11.8 - 6.82i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.6 + 7.87i)T + (48.5 + 84.0i)T^{2} \)
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

−9.314948655938764792242746447545, −8.532703915154426149286106725687, −7.68086800340430714067243857302, −7.01325845246787086735378480690, −5.64867622653524535535974085454, −5.45268479039523484826597650689, −4.23145666994468482497412375535, −3.20874304054601242635420098588, −1.43126369505426568373023990434, −0.72909978330410436089991635653, 1.37453479662564469203960515817, 2.66271171146598301142348485348, 3.35562988999185769244356928405, 4.22086712322484745512391099487, 5.97813890218754790106164786395, 6.35662885835335630367826242702, 7.25133300925603191064400982699, 7.941596651443437511332427770799, 8.920487508780303572038154233811, 9.624419856065122226707240311150

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