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Properties

Label 2-1860-1860.959-c0-0-1
Degree $2$
Conductor $1860$
Sign $0.569 + 0.822i$
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  + (−0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s − 5-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 − 0.587i)10-s + (−0.309 − 0.951i)12-s + (−0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)18-s + (0.690 − 0.951i)19-s + (−0.309 + 0.951i)20-s + (0.190 − 0.587i)23-s + (0.809 + 0.587i)24-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s − 5-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 − 0.587i)10-s + (−0.309 − 0.951i)12-s + (−0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)18-s + (0.690 − 0.951i)19-s + (−0.309 + 0.951i)20-s + (0.190 − 0.587i)23-s + (0.809 + 0.587i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ 0.569 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8118646553\)
\(L(\frac12)\) \(\approx\) \(0.8118646553\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + T \)
31 \( 1 + (0.809 + 0.587i)T \)
good7 \( 1 + (-0.809 + 0.587i)T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 - 1.17iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−8.907706102515296388969169680520, −8.593651513844829349233608557951, −7.67107302251207786555335136624, −7.21349042034613693592827418804, −6.57621769153610561090583422930, −5.42868939354976063403352338767, −4.35711757772140610690028697234, −3.24364145005132855106014252961, −2.18009902994333449570581966654, −0.76428550921295347572565042270, 1.49575708439865911393809218343, 2.81769227945297205556446243812, 3.55078957551036625109132315300, 4.19117222202995620490057919495, 5.28894540769403650443196585783, 6.80625363589277479611478844496, 7.66223298685654227327123737123, 8.023478202977722342807222608268, 8.860035313651529217537095959345, 9.473278550953855459395229689402

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