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Properties

Label 2-1140-1140.179-c0-0-3
Degree $2$
Conductor $1140$
Sign $-0.813 + 0.582i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s + (0.499 + 0.866i)12-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)20-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s + (0.499 + 0.866i)12-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ -0.813 + 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9606098876\)
\(L(\frac12)\) \(\approx\) \(0.9606098876\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.73iT - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.703087537625529336268491072321, −8.807747836223584929844541887408, −8.280766370437092961227546576922, −7.53191296207671711697830257977, −6.46753058674547840635102946477, −5.34900201534526514661965425328, −4.22813377514899837428763554232, −3.06760572592867028226998571665, −2.05559530897385036239021788498, −1.03872888843775910444381660840, 2.02258717077329437599779184927, 3.28014588770826760071059792705, 4.41222342364458943284308456279, 5.36135692094278045153549721214, 6.23427632194586153163138489735, 7.02134679169361115077793797484, 8.039617471650356980984865039318, 8.642044085026649807091077190094, 9.592605708246246520154560015035, 10.08843894636693517243216363931

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