↑

Properties

Label 2-2268-63.58-c1-0-15
Degree $2$
Conductor $2268$
Sign $0.984 - 0.175i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + (0.951 + 1.64i)5-s + (1.46 − 2.20i)7-s + (1.41 − 2.44i)11-s + (−2.41 + 4.17i)13-s + (2.14 + 3.70i)17-s + (2.37 − 4.11i)19-s + (−1.23 − 2.13i)23-s + (0.689 − 1.19i)25-s + (4.32 + 7.49i)29-s + 3.37·31-s + (5.02 + 0.310i)35-s + (−2.59 + 4.48i)37-s + (4.10 − 7.10i)41-s + (−3.36 − 5.82i)43-s − 1.72·47-s + ⋯
L(s)  = 1  + (0.425 + 0.737i)5-s + (0.552 − 0.833i)7-s + (0.426 − 0.737i)11-s + (−0.669 + 1.15i)13-s + (0.519 + 0.899i)17-s + (0.544 − 0.943i)19-s + (−0.257 − 0.445i)23-s + (0.137 − 0.238i)25-s + (0.803 + 1.39i)29-s + 0.605·31-s + (0.849 + 0.0524i)35-s + (−0.426 + 0.738i)37-s + (0.640 − 1.10i)41-s + (−0.513 − 0.888i)43-s − 0.252·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.984 - 0.175i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.984 - 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.166632933\)
\(L(\frac12)\) \(\approx\) \(2.166632933\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-1.46 + 2.20i)T \)
good5 \( 1 + (-0.951 - 1.64i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.41 + 2.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.41 - 4.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.14 - 3.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.37 + 4.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 + 2.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.32 - 7.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 + (2.59 - 4.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.10 + 7.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.36 + 5.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.72T + 47T^{2} \)
53 \( 1 + (-5.80 - 10.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 5.24T + 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 - 7.39T + 71T^{2} \)
73 \( 1 + (-4.23 - 7.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.94T + 79T^{2} \)
83 \( 1 + (4.72 + 8.18i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.91 + 8.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.60 + 2.78i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.901729567504081259678534996568, −8.417722114512323019476893958774, −7.18761981966133764793543714016, −6.91377167933112909439277254467, −6.03981547841998255324390525346, −5.00187561612200665437054760214, −4.18647696149365041372016020510, −3.25408633007624470159924726668, −2.20129936069426071403403384480, −1.02796936047332067314023437245, 0.987803480648340286507113111412, 2.09609242063711890105776046672, 3.04730537932945037332534801937, 4.33581674918858203281396925755, 5.25629773253001417320410011595, 5.52074047324691560610124682552, 6.61803195380702098853420950401, 7.76343127544804852980197116980, 8.082291978792224891576778188094, 9.084842380959528983212730816130

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