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Properties

Label 1-1881-1881.520-r0-0-0
Degree $1$
Conductor $1881$
Sign $-0.737 - 0.674i$
Analytic cond. $8.73532$
Root an. cond. $8.73532$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.669 − 0.743i)5-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.309 − 0.951i)13-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)16-s + (−0.978 − 0.207i)17-s + (−0.978 + 0.207i)20-s + 23-s + (−0.104 − 0.994i)25-s + (−0.809 − 0.587i)26-s + (0.669 − 0.743i)28-s + (0.913 − 0.406i)29-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.669 − 0.743i)5-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.309 − 0.951i)13-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)16-s + (−0.978 − 0.207i)17-s + (−0.978 + 0.207i)20-s + 23-s + (−0.104 − 0.994i)25-s + (−0.809 − 0.587i)26-s + (0.669 − 0.743i)28-s + (0.913 − 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1881\)    =    \(3^{2} \cdot 11 \cdot 19\)
Sign: $-0.737 - 0.674i$
Analytic conductor: \(8.73532\)
Root analytic conductor: \(8.73532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1881} (520, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1881,\ (0:\ ),\ -0.737 - 0.674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6371221950 - 1.640803936i\)
\(L(\frac12)\) \(\approx\) \(0.6371221950 - 1.640803936i\)
\(L(1)\) \(\approx\) \(0.9720470977 - 0.7666592029i\)
\(L(1)\) \(\approx\) \(0.9720470977 - 0.7666592029i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.669 - 0.743i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−20.66668127413214388837150330470, −19.35170827951503030881977323400, −18.86997865483637655635937824456, −17.85284809210020134044791955112, −17.383070756365678804023866320542, −16.80326895101046323286946469443, −15.900658408868405422227011368712, −15.26536622324251318119801352405, −14.22949029384540667811931104646, −13.99295553862088432238719519448, −13.27425506270354025487676168492, −12.5381458288297847575894954177, −11.2818359915394662071633967044, −10.702618944879254848811973213823, −9.67687994218874429118382864356, −9.11008718026032636903602996412, −8.11434293158498593689258252548, −7.188658456102378107083388679909, −6.62606490581524823848295513676, −6.1653215780407087989450848248, −4.96631904713934878590187633868, −4.26937332734408465396881157462, −3.41872826991266397052265326082, −2.44948882002466445219029282808, −1.11698107665079289066053420222, 0.62212479138919309468617111797, 1.64250650424935574101473531683, 2.52895926392310363300731285210, 3.15038868969833266062006941856, 4.448551179662961567358573532620, 5.0617885970400766622107787161, 5.78686348986726375704689250614, 6.48581478483486403587264068053, 8.03234030849268338357018288328, 8.9261523089636497198331512941, 9.1367968154391526608152221370, 10.1914414618405418975834090607, 10.81804283472808669233138912204, 11.78335305283209589173156865612, 12.46210536302185394677855269759, 13.007262742566035380115250401858, 13.64802988694165270466696898374, 14.47025946697748526844660826671, 15.43891639486309987824320256733, 15.90995569003567820172273637810, 17.22614040794124151098128066415, 17.781187379060681618568849998436, 18.314157653024145122437077311, 19.31007666054257875481832737399, 19.829401100943251220078300376349

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