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Properties

Label 1-1840-1840.1413-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.978 - 0.205i$
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  + (0.415 + 0.909i)3-s + (0.540 − 0.841i)7-s + (−0.654 + 0.755i)9-s + (−0.989 + 0.142i)11-s + (0.841 − 0.540i)13-s + (−0.281 − 0.959i)17-s + (−0.281 + 0.959i)19-s + (0.989 + 0.142i)21-s + (−0.959 − 0.281i)27-s + (−0.281 − 0.959i)29-s + (0.415 − 0.909i)31-s + (−0.540 − 0.841i)33-s + (0.654 − 0.755i)37-s + (0.841 + 0.540i)39-s + (0.654 + 0.755i)41-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)3-s + (0.540 − 0.841i)7-s + (−0.654 + 0.755i)9-s + (−0.989 + 0.142i)11-s + (0.841 − 0.540i)13-s + (−0.281 − 0.959i)17-s + (−0.281 + 0.959i)19-s + (0.989 + 0.142i)21-s + (−0.959 − 0.281i)27-s + (−0.281 − 0.959i)29-s + (0.415 − 0.909i)31-s + (−0.540 − 0.841i)33-s + (0.654 − 0.755i)37-s + (0.841 + 0.540i)39-s + (0.654 + 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.978 - 0.205i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.978 - 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.728347250 - 0.1797658092i\)
\(L(\frac12)\) \(\approx\) \(1.728347250 - 0.1797658092i\)
\(L(1)\) \(\approx\) \(1.208936135 + 0.1257546644i\)
\(L(1)\) \(\approx\) \(1.208936135 + 0.1257546644i\)

Euler product

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

−20.11484991928951810024804921574, −19.217908880132020574600831177, −18.78046686082712673336691848201, −17.94379186061876878336360427558, −17.64760017140348982411910012653, −16.491558663601777672328690340807, −15.53679944751847245576348497370, −15.05330417603512616955320302490, −14.14311957750815765374353187181, −13.47597521530502357915054783579, −12.756941496981217506240184552356, −12.156387353624731282980180745437, −11.18346594691924824134779970830, −10.696801320224776669017279787123, −9.307117488292915451596353789948, −8.63256802525420805197936920689, −8.2119204900542301319305606371, −7.25367634865623962088441725438, −6.41611068785023587700254560156, −5.70085064466527172581822129618, −4.803084943272324787557292023648, −3.638793148303808685992572311158, −2.651078719794641751599591993529, −2.022132789686154434569818272404, −1.06087165636120999077511807092, 0.643633705959028940123116028470, 2.068336180313729567142276944739, 2.91213255630940408207000741089, 3.90735216332647834754629468347, 4.47081357168399718590842680824, 5.36762079194510082300498469458, 6.14274599029630080217828204125, 7.59165009897763476080862098055, 7.88863243025073473965157066167, 8.76370474525182341308577734904, 9.78069320356310840283413066420, 10.291882335695903452922336748153, 11.04703005702487089421765695047, 11.600644498624440424266364585318, 13.01979946821848532242313152049, 13.47544520789522634310351899789, 14.29064495509922189318697943607, 14.97311063226891524533932331493, 15.74611966508679228675041382458, 16.32310964298846734820131582541, 17.058617625417774447254408650206, 17.94896639549203452729838934960, 18.57431117088190506758561050064, 19.614994547943639891141098890476, 20.35105256314033170072381234176

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