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Properties

Label 2-1860-1860.1199-c0-0-0
Degree $2$
Conductor $1860$
Sign $0.306 - 0.951i$
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  + (−0.669 − 0.743i)2-s + (0.104 + 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.5 − 0.866i)5-s + (0.669 − 0.743i)6-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.309 + 0.951i)10-s − 12-s + (0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (−1.28 + 1.15i)17-s + (0.809 + 0.587i)18-s + (0.809 + 1.81i)19-s + (0.913 − 0.406i)20-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (0.104 + 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.5 − 0.866i)5-s + (0.669 − 0.743i)6-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.309 + 0.951i)10-s − 12-s + (0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (−1.28 + 1.15i)17-s + (0.809 + 0.587i)18-s + (0.809 + 1.81i)19-s + (0.913 − 0.406i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.306 - 0.951i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ 0.306 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5646673578\)
\(L(\frac12)\) \(\approx\) \(0.5646673578\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.669 + 0.743i)T \)
3 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
good7 \( 1 + (0.913 + 0.406i)T^{2} \)
11 \( 1 + (0.104 + 0.994i)T^{2} \)
13 \( 1 + (0.669 + 0.743i)T^{2} \)
17 \( 1 + (1.28 - 1.15i)T + (0.104 - 0.994i)T^{2} \)
19 \( 1 + (-0.809 - 1.81i)T + (-0.669 + 0.743i)T^{2} \)
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 \( 1 + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.669 + 0.743i)T^{2} \)
47 \( 1 + (-0.873 - 1.20i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.244 - 1.14i)T + (-0.913 + 0.406i)T^{2} \)
59 \( 1 + (-0.978 + 0.207i)T^{2} \)
61 \( 1 - 0.813iT - T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.913 - 0.406i)T^{2} \)
73 \( 1 + (-0.104 - 0.994i)T^{2} \)
79 \( 1 + (0.139 + 0.155i)T + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (0.139 - 1.33i)T + (-0.978 - 0.207i)T^{2} \)
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

−9.531278960561585526944561189588, −8.905553066207258791410556809271, −8.276172256073029016846940760542, −7.76313628149800447010315787551, −6.39638899975304141476092151811, −5.29472506083370788595303985261, −4.26749344337508589362087704428, −3.90076033509509962191371485772, −2.78485343160273729563531590246, −1.43880888552142799686607122879, 0.53547809248864380975729817876, 2.18930938288669689173225595178, 3.02050184437662313413975530613, 4.56972735172502102765311420388, 5.49695614201242052303271989910, 6.61435390391990584718021303989, 6.93339872949936544071789602558, 7.51914548267199825200266921669, 8.331785328484074787462818182201, 9.108652173234787823726141479918

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