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Properties

Label 2-1860-1860.239-c0-0-2
Degree $2$
Conductor $1860$
Sign $0.542 - 0.840i$
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  + (0.104 + 0.994i)2-s + (0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.5 − 0.866i)5-s + (−0.104 + 0.994i)6-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (0.809 − 0.587i)10-s − 12-s + (−0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (1.72 − 0.181i)17-s + (−0.309 + 0.951i)18-s + (−0.309 + 0.278i)19-s + (0.669 + 0.743i)20-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.5 − 0.866i)5-s + (−0.104 + 0.994i)6-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (0.809 − 0.587i)10-s − 12-s + (−0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (1.72 − 0.181i)17-s + (−0.309 + 0.951i)18-s + (−0.309 + 0.278i)19-s + (0.669 + 0.743i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.542 - 0.840i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ 0.542 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.474384351\)
\(L(\frac12)\) \(\approx\) \(1.474384351\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
good7 \( 1 + (0.669 - 0.743i)T^{2} \)
11 \( 1 + (0.978 + 0.207i)T^{2} \)
13 \( 1 + (-0.104 - 0.994i)T^{2} \)
17 \( 1 + (-1.72 + 0.181i)T + (0.978 - 0.207i)T^{2} \)
19 \( 1 + (0.309 - 0.278i)T + (0.104 - 0.994i)T^{2} \)
23 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.913 + 0.406i)T^{2} \)
43 \( 1 + (0.104 - 0.994i)T^{2} \)
47 \( 1 + (1.89 - 0.614i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.773 + 1.73i)T + (-0.669 - 0.743i)T^{2} \)
59 \( 1 + (0.913 + 0.406i)T^{2} \)
61 \( 1 + 1.48iT - T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.669 + 0.743i)T^{2} \)
73 \( 1 + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (-0.204 - 1.94i)T + (-0.978 + 0.207i)T^{2} \)
83 \( 1 + (-0.204 + 0.0434i)T + (0.913 - 0.406i)T^{2} \)
89 \( 1 + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.405039264720477378186227035205, −8.388876180182083127164635456471, −8.220562284729220286559253653924, −7.35978892410379753285560010682, −6.57659544170851020067101371027, −5.26251385741106786361293990428, −4.82951130203263623367815236888, −3.79817689654692047996910492836, −3.11076988479234764192075042546, −1.26914450549733280492875880997, 1.34555786771373172002042354593, 2.56275831012418103526700752347, 3.30848149815427826774253009639, 3.85060939696296299778299144302, 4.97783356406954719319074414463, 6.09610973938716601785352033802, 7.26764922109215074390306584815, 7.79733986827038469746924849312, 8.616260165492055492669241365823, 9.443652817108798842076818757093

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