↑

Properties

Label 2-1140-1140.1139-c0-0-18
Degree $2$
Conductor $1140$
Sign $-0.923 - 0.382i$
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  + (0.382 − 0.923i)2-s + (−0.923 + 0.382i)3-s + (−0.707 − 0.707i)4-s − i·5-s + i·6-s + (−0.923 + 0.382i)8-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)10-s − 1.41·11-s + (0.923 + 0.382i)12-s − 0.765·13-s + (0.382 + 0.923i)15-s + i·16-s + (−0.382 − 0.923i)18-s − i·19-s + (−0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.923 + 0.382i)3-s + (−0.707 − 0.707i)4-s − i·5-s + i·6-s + (−0.923 + 0.382i)8-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)10-s − 1.41·11-s + (0.923 + 0.382i)12-s − 0.765·13-s + (0.382 + 0.923i)15-s + i·16-s + (−0.382 − 0.923i)18-s − i·19-s + (−0.707 + 0.707i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.923 - 0.382i$
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: \((2,\ 1140,\ (\ :0),\ -0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4259327768\)
\(L(\frac12)\) \(\approx\) \(0.4259327768\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.801141542962649054643459878402, −9.059110783748855464010625784421, −8.103260596206820702483632581149, −6.87891115330937498359513414403, −5.64101951648516251078617287110, −5.04567866818263802460170250305, −4.59501529660371989249966404328, −3.34121482335924554409084126337, −1.98335054218124991027923072276, −0.35787527379877901279590993896, 2.35055181698212620220721623736, 3.56499262668969179920372019647, 4.82716869072818226712161262307, 5.52374759140254208738821817404, 6.26445653493603801675341614648, 7.18595951809070138429494464088, 7.57877677511314777714753086850, 8.453726091473655477859445744576, 9.955633201894699214184864499250, 10.31202952478420634953033464124

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