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Properties

Label 1-403-403.379-r1-0-0
Degree $1$
Conductor $403$
Sign $0.428 + 0.903i$
Analytic cond. $43.3083$
Root an. cond. $43.3083$
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  + (−0.994 + 0.104i)2-s + (−0.104 + 0.994i)3-s + (0.978 − 0.207i)4-s + (−0.866 + 0.5i)5-s − i·6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.978 − 0.207i)9-s + (0.809 − 0.587i)10-s + (0.951 + 0.309i)11-s + (0.104 + 0.994i)12-s + (−0.978 − 0.207i)14-s + (−0.406 − 0.913i)15-s + (0.913 − 0.406i)16-s + (−0.309 − 0.951i)17-s + (0.994 + 0.104i)18-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (−0.104 + 0.994i)3-s + (0.978 − 0.207i)4-s + (−0.866 + 0.5i)5-s − i·6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.978 − 0.207i)9-s + (0.809 − 0.587i)10-s + (0.951 + 0.309i)11-s + (0.104 + 0.994i)12-s + (−0.978 − 0.207i)14-s + (−0.406 − 0.913i)15-s + (0.913 − 0.406i)16-s + (−0.309 − 0.951i)17-s + (0.994 + 0.104i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.428 + 0.903i$
Analytic conductor: \(43.3083\)
Root analytic conductor: \(43.3083\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (1:\ ),\ 0.428 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.032386989 + 0.6532737026i\)
\(L(\frac12)\) \(\approx\) \(1.032386989 + 0.6532737026i\)
\(L(1)\) \(\approx\) \(0.6803716551 + 0.3026928974i\)
\(L(1)\) \(\approx\) \(0.6803716551 + 0.3026928974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.994 + 0.104i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (0.978 + 0.207i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (0.587 - 0.809i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.207 + 0.978i)T \)
73 \( 1 + (-0.207 - 0.978i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.406 - 0.913i)T \)
89 \( 1 + (-0.743 + 0.669i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−24.20464372919850554484758710368, −23.58781672646155067756911680667, −22.28753076598638166112717858885, −21.04458034902157633621134437780, −20.0569903655468663540560881872, −19.62467384029684474701331251725, −18.8163624043854234839193664234, −17.827911396006567448975705559381, −17.15038695562332638041987439620, −16.48278949596186743251994649080, −15.22534370033280090135292104789, −14.37261979285638132171079742670, −13.027264282848288833113939046119, −12.13351180120843690372031159510, −11.34269823631926476630966588739, −10.84771580787991712829915878894, −9.00846013745998141265288384452, −8.56320278421177240148255932393, −7.543751579624656756118281203136, −6.9798161870114758569901942358, −5.71148358400197143343683816048, −4.23516761361851254074349698778, −2.83039354195083245706056348391, −1.3619068048669060915987632360, −0.85730438396119572943741554836, 0.726131194058773262814071085895, 2.36329024309989279479517264788, 3.55910098146416403995065306774, 4.681805103715281576506189322, 5.89271452326980155544978511929, 7.13482898498941447145641638905, 8.01124041568928592714798964874, 8.97386598138100345395100526251, 9.738970892291553566294751287454, 10.874740573327685573107670500798, 11.50228462610085468701784577233, 12.01730477198409002712409167385, 14.25018094193538610460947471823, 14.86266471837180358487936162710, 15.56488749606110199781062001153, 16.40028733194529424811455375883, 17.28234279301922171570076021081, 18.11689731271638170492775763885, 19.0330495470332752540875614089, 20.017257458803935880592172086554, 20.61273146077495447542387681144, 21.528912980794444742839482494554, 22.59611272201724588558916194252, 23.34204806118768218900304615540, 24.62329348627359231690129726780

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