Properties

Label 2-1860-1860.719-c0-0-3
Degree $2$
Conductor $1860$
Sign $-0.970 + 0.242i$
Analytic cond. $0.928260$
Root an. cond. $0.963462$
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 + (−0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s + 0.999·12-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (1.5 − 0.866i)17-s + 0.999·18-s + (1.5 − 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 + (−0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s + 0.999·12-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (1.5 − 0.866i)17-s + 0.999·18-s + (1.5 − 0.866i)19-s + (0.499 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.970 + 0.242i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ -0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7335132336\)
\(L(\frac12)\) \(\approx\) \(0.7335132336\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + 2T + T^{2} \)
29 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + 1.73iT - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 - 1.73iT - T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.200623121079635895436243915718, −8.280470741270518240845199245697, −7.67085084270521232180187820740, −6.93564146566920443849925101270, −5.51745900886662951846251860828, −5.26777616187360311551577507386, −3.95156007953557878507315166732, −2.73156890467627738618637121648, −1.72939569467481382069176179786, −0.74359213069873547407954506489, 1.56923718534117202579117379907, 3.28882891341080085674290845811, 4.08162247933992172967937618909, 5.38008321369507625796634825797, 5.82802862488768338832935354203, 6.38356022435968803072346121589, 7.58534180535230040184065763639, 7.999766181447529397573491630855, 9.293895101378969554992896468928, 9.760719117157179335477536961089

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