↑

Properties

Label 2-1140-1140.1139-c0-0-2
Degree $2$
Conductor $1140$
Sign $-0.382 - 0.923i$
Analytic cond. $0.568934$
Root an. cond. $0.754277$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 + 0.382i)2-s + (−0.382 + 0.923i)3-s + (0.707 − 0.707i)4-s + i·5-s − i·6-s + (−0.382 + 0.923i)8-s + (−0.707 − 0.707i)9-s + (−0.382 − 0.923i)10-s + 1.41·11-s + (0.382 + 0.923i)12-s + 1.84·13-s + (−0.923 − 0.382i)15-s − i·16-s + (0.923 + 0.382i)18-s + i·19-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + (−0.382 + 0.923i)3-s + (0.707 − 0.707i)4-s + i·5-s − i·6-s + (−0.382 + 0.923i)8-s + (−0.707 − 0.707i)9-s + (−0.382 − 0.923i)10-s + 1.41·11-s + (0.382 + 0.923i)12-s + 1.84·13-s + (−0.923 − 0.382i)15-s − i·16-s + (0.923 + 0.382i)18-s + i·19-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6800614618\)
\(L(\frac12)\) \(\approx\) \(0.6800614618\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 - iT \)
19 \( 1 - iT \)
good7 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - 1.84T + T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 0.765T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.84iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 - 0.765iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 0.765T + T^{2} \)
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

−10.24903549325416445013424347846, −9.471413672185429388827894198670, −8.766651029055379223690320304403, −7.987215504568555716927691009489, −6.64471705603235055675335402222, −6.34894162172834793327355081694, −5.51066518853629450632785097596, −3.97844817805381805216106363766, −3.28431726551207666395881967294, −1.54908888480725673976834184362, 1.02823643925373005346255515072, 1.72620469047155248410475309096, 3.31336187800640273452139889801, 4.44522421067357342161051062425, 5.90128878415812979562929836851, 6.48321599819742778840492460927, 7.38284042351120628395148565166, 8.386592761203111384883121999203, 8.817733986735110920187485265477, 9.470978098584229981115950913633

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