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Properties

Label 2-1860-1860.1739-c0-0-0
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: $\chi_{1860} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ -0.502 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1788552695\)
\(L(\frac12)\) \(\approx\) \(0.1788552695\)
\(L(1)\) 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

−9.924267165981775043788579657927, −9.231163589317516359730651344654, −8.386288989404681841299217978038, −7.80477568997620363453204982694, −6.71667041464488973276965777957, −6.24963526019567963896565299835, −5.18226065538421033651474893366, −4.22274063075785618778591202846, −3.18200326768001394979530522853, −2.14794387189691229536480369948, 0.17061751165116463307365468711, 1.57812816497405898210592485054, 2.42532155597575314965973997676, 3.71511538721632008553068464466, 4.73311860498368002482672667757, 5.98141784175663587316031829130, 6.80289771975937942633769379933, 7.45286020565976588980908538278, 8.331633173792443390080683385493, 8.752043450444746173874533067456

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