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Properties

Label 2-1860-1860.1199-c0-0-2
Degree $2$
Conductor $1860$
Sign $-0.502 - 0.864i$
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  + (0.913 + 0.406i)2-s + (0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + (0.5 + 0.866i)5-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.104 + 0.994i)10-s + (−0.669 + 0.743i)12-s + (−0.809 + 0.587i)15-s + (−0.104 + 0.994i)16-s + (1.28 − 1.15i)17-s + (−0.978 − 0.207i)18-s + (−0.809 − 1.81i)19-s + (−0.309 + 0.951i)20-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + (0.5 + 0.866i)5-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.104 + 0.994i)10-s + (−0.669 + 0.743i)12-s + (−0.809 + 0.587i)15-s + (−0.104 + 0.994i)16-s + (1.28 − 1.15i)17-s + (−0.978 − 0.207i)18-s + (−0.809 − 1.81i)19-s + (−0.309 + 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.502 - 0.864i$
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: \((2,\ 1860,\ (\ :0),\ -0.502 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.160074588\)
\(L(\frac12)\) \(\approx\) \(2.160074588\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.913 - 0.406i)T \)
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

−9.667994611977029646804957689372, −9.019996867448253897054349615216, −7.976837582802114970164358118389, −7.12637273221822651165663245178, −6.41365438832127223662136459989, −5.49633354887334351383473388770, −4.90499096507938091133795221187, −3.94173378842341594301041598851, −2.97957180213601084421795797664, −2.44924664130150271713072593523, 1.39168437107874467355329039400, 1.88641005498357129818911909852, 3.25968842984587021400459646163, 4.08366887893652301688017258527, 5.38369649213999038657207478648, 5.79714236523696041186868227553, 6.48927764254152569556837746293, 7.61394041385909770071204389597, 8.228865174339714994616084457887, 9.170922571533276151383834201758

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