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Properties

Label 1-1840-1840.1803-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.514 + 0.857i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.142 − 0.989i)3-s + (−0.755 + 0.654i)7-s + (−0.959 − 0.281i)9-s + (0.540 − 0.841i)11-s + (−0.654 + 0.755i)13-s + (0.909 − 0.415i)17-s + (−0.909 − 0.415i)19-s + (0.540 + 0.841i)21-s + (−0.415 + 0.909i)27-s + (−0.909 + 0.415i)29-s + (0.142 + 0.989i)31-s + (−0.755 − 0.654i)33-s + (−0.959 − 0.281i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)3-s + (−0.755 + 0.654i)7-s + (−0.959 − 0.281i)9-s + (0.540 − 0.841i)11-s + (−0.654 + 0.755i)13-s + (0.909 − 0.415i)17-s + (−0.909 − 0.415i)19-s + (0.540 + 0.841i)21-s + (−0.415 + 0.909i)27-s + (−0.909 + 0.415i)29-s + (0.142 + 0.989i)31-s + (−0.755 − 0.654i)33-s + (−0.959 − 0.281i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6670276646 + 0.3774331471i\)
\(L(\frac12)\) \(\approx\) \(0.6670276646 + 0.3774331471i\)
\(L(1)\) \(\approx\) \(0.8478499374 - 0.1465151841i\)
\(L(1)\) \(\approx\) \(0.8478499374 - 0.1465151841i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (-0.755 + 0.654i)T \)
11 \( 1 + (0.540 - 0.841i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.909 - 0.415i)T \)
29 \( 1 + (-0.909 + 0.415i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.755 - 0.654i)T \)
61 \( 1 + (-0.989 + 0.142i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.13991528599895305708264743410, −19.45643421899040184329808683100, −18.767329568671818583972938438152, −17.41768754286815756193650045423, −17.03314155091312618210258534923, −16.49346741811710853577244972685, −15.46124987143560783192805359233, −14.95137372813765181959020653239, −14.338384670995488316994921514178, −13.36490146304569827149433747191, −12.578542811830569917312656944982, −11.84767061346686521766096087247, −10.77410564390430564611405612934, −10.07063302169979293881748631815, −9.78572414097445438069927680816, −8.8346873344245066288338450654, −7.882617805779427954548761940661, −7.1351521675532692015894371871, −6.08299358069214185920084355085, −5.35913524785295163354875409394, −4.27437647767983463988815544246, −3.81122243694209850088705243621, −2.91813425408170396356464127357, −1.865624980640104334116180818976, −0.28360770535151335588409107163, 1.0545814829176332700728725009, 2.0977405676707534669823397182, 2.91461839343750523726433102133, 3.65769910706991443928328481238, 4.990281107642928610720524134873, 5.91475650062759125894884312744, 6.50667582227110008287212713290, 7.22650515835757850181558518965, 8.11298620814298454662217905787, 9.103258544331057399889256388334, 9.30941940386703414430464139885, 10.67338169818659787170996528420, 11.49408337933313874528116502089, 12.30177745224451234773353477158, 12.62802008393952043905725999908, 13.65255417815611641391255903692, 14.24698938253337326554308732703, 14.905130260271930195232637933237, 15.98657498490833881858189332732, 16.688561854767163383473671404187, 17.2980640300813854632151604205, 18.29271210536721858077473301823, 18.92486242282403120578298410023, 19.36890885954851898388805728333, 19.90768886806256994222024588001

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