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Properties

Label 2-1860-1860.539-c0-0-2
Degree $2$
Conductor $1860$
Sign $0.999 + 0.0325i$
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  + (−0.978 + 0.207i)2-s + (0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (0.5 − 0.866i)5-s + (−0.978 − 0.207i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.309 + 0.951i)10-s + 1.00·12-s + (0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (−0.360 − 1.69i)17-s + (−0.809 − 0.587i)18-s + (0.809 + 0.0850i)19-s + (0.104 − 0.994i)20-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (0.5 − 0.866i)5-s + (−0.978 − 0.207i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.309 + 0.951i)10-s + 1.00·12-s + (0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (−0.360 − 1.69i)17-s + (−0.809 − 0.587i)18-s + (0.809 + 0.0850i)19-s + (0.104 − 0.994i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.999 + 0.0325i$
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: \((2,\ 1860,\ (\ :0),\ 0.999 + 0.0325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.148073870\)
\(L(\frac12)\) \(\approx\) \(1.148073870\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.978 - 0.207i)T \)
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

−9.402860782501818862369420583950, −8.828597098904265257208520275595, −7.979075748410188394202662079336, −7.41777275555163210364997094704, −6.43578520506366531701263010641, −5.30851105961601032141776308474, −4.69534924454729604473703186627, −3.25041961733510685430082828052, −2.35190921652013615268389062571, −1.23513998674002372905523014041, 1.49162895525118583810608517750, 2.36140900974164937430428887430, 3.17536929976691980073960732125, 4.07511632769051389709463098505, 5.87484575897641766256686990105, 6.54449670324597631312889614226, 7.20429907077334856162794604561, 8.044005885226454357978565251088, 8.570318303445721247197361376039, 9.468312885731899430048091565541

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