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Properties

Label 2-1860-1860.539-c0-0-3
Degree $2$
Conductor $1860$
Sign $-0.970 - 0.239i$
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 + (−0.913 − 0.406i)3-s + (−0.978 + 0.207i)4-s + (0.5 − 0.866i)5-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (0.669 + 0.743i)9-s + (−0.913 − 0.406i)10-s + (0.978 + 0.207i)12-s + (−0.809 + 0.587i)15-s + (0.913 − 0.406i)16-s + (−0.360 − 1.69i)17-s + (0.669 − 0.743i)18-s + (−0.809 − 0.0850i)19-s + (−0.309 + 0.951i)20-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.913 − 0.406i)3-s + (−0.978 + 0.207i)4-s + (0.5 − 0.866i)5-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (0.669 + 0.743i)9-s + (−0.913 − 0.406i)10-s + (0.978 + 0.207i)12-s + (−0.809 + 0.587i)15-s + (0.913 − 0.406i)16-s + (−0.360 − 1.69i)17-s + (0.669 − 0.743i)18-s + (−0.809 − 0.0850i)19-s + (−0.309 + 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.970 - 0.239i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ -0.970 - 0.239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5957176542\)
\(L(\frac12)\) \(\approx\) \(0.5957176542\)
\(L(1)\) 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 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
good7 \( 1 + (-0.104 - 0.994i)T^{2} \)
11 \( 1 + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (-0.978 + 0.207i)T^{2} \)
17 \( 1 + (0.360 + 1.69i)T + (-0.913 + 0.406i)T^{2} \)
19 \( 1 + (0.809 + 0.0850i)T + (0.978 + 0.207i)T^{2} \)
23 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.669 + 0.743i)T^{2} \)
43 \( 1 + (0.978 + 0.207i)T^{2} \)
47 \( 1 + (-0.244 - 0.336i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.873 - 0.786i)T + (0.104 - 0.994i)T^{2} \)
59 \( 1 + (0.669 + 0.743i)T^{2} \)
61 \( 1 + 1.98iT - T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.104 + 0.994i)T^{2} \)
73 \( 1 + (0.913 + 0.406i)T^{2} \)
79 \( 1 + (-1.78 + 0.379i)T + (0.913 - 0.406i)T^{2} \)
83 \( 1 + (1.78 - 0.795i)T + (0.669 - 0.743i)T^{2} \)
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

−9.326452799774879695306745454653, −8.379845526555355119219936094349, −7.54281523942132736280313354115, −6.49513693483728468374247293214, −5.55088852585989883200940232312, −4.84114604486889142067433990549, −4.24958655562005437613727248894, −2.67966576657242723947674565942, −1.74339376661423813286234761720, −0.52697518935719551352472479972, 1.69918517546968472524704118745, 3.51833650549933272186680368404, 4.23724151997532187477016262934, 5.34774169973252383278473423015, 5.92987360956563360380247952246, 6.61000973686776264126190754407, 7.16542345519925080028775594197, 8.241719368510555143595555036509, 9.073387547125364658414613016907, 9.881637699536531717531515171587

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