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Properties

Label 1-283-283.180-r1-0-0
Degree $1$
Conductor $283$
Sign $0.228 - 0.973i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (0.296 − 0.955i)2-s + (0.970 − 0.242i)3-s + (−0.824 − 0.565i)4-s + (0.848 − 0.528i)5-s + (0.0556 − 0.998i)6-s + (0.556 + 0.830i)7-s + (−0.784 + 0.619i)8-s + (0.882 − 0.470i)9-s + (−0.253 − 0.967i)10-s + (0.902 + 0.431i)11-s + (−0.937 − 0.348i)12-s + (0.836 + 0.547i)13-s + (0.958 − 0.285i)14-s + (0.695 − 0.718i)15-s + (0.359 + 0.933i)16-s + (−0.726 + 0.687i)17-s + ⋯
L(s)  = 1  + (0.296 − 0.955i)2-s + (0.970 − 0.242i)3-s + (−0.824 − 0.565i)4-s + (0.848 − 0.528i)5-s + (0.0556 − 0.998i)6-s + (0.556 + 0.830i)7-s + (−0.784 + 0.619i)8-s + (0.882 − 0.470i)9-s + (−0.253 − 0.967i)10-s + (0.902 + 0.431i)11-s + (−0.937 − 0.348i)12-s + (0.836 + 0.547i)13-s + (0.958 − 0.285i)14-s + (0.695 − 0.718i)15-s + (0.359 + 0.933i)16-s + (−0.726 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.228 - 0.973i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.228 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.204786850 - 2.540703178i\)
\(L(\frac12)\) \(\approx\) \(3.204786850 - 2.540703178i\)
\(L(1)\) \(\approx\) \(1.816204217 - 1.059898340i\)
\(L(1)\) \(\approx\) \(1.816204217 - 1.059898340i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.296 - 0.955i)T \)
3 \( 1 + (0.970 - 0.242i)T \)
5 \( 1 + (0.848 - 0.528i)T \)
7 \( 1 + (0.556 + 0.830i)T \)
11 \( 1 + (0.902 + 0.431i)T \)
13 \( 1 + (0.836 + 0.547i)T \)
17 \( 1 + (-0.726 + 0.687i)T \)
19 \( 1 + (0.892 - 0.451i)T \)
23 \( 1 + (0.811 + 0.584i)T \)
29 \( 1 + (-0.944 + 0.328i)T \)
31 \( 1 + (0.460 + 0.887i)T \)
37 \( 1 + (-0.317 - 0.948i)T \)
41 \( 1 + (-0.929 + 0.369i)T \)
43 \( 1 + (-0.920 - 0.390i)T \)
47 \( 1 + (-0.0556 - 0.998i)T \)
53 \( 1 + (-0.359 + 0.933i)T \)
59 \( 1 + (-0.610 - 0.791i)T \)
61 \( 1 + (0.964 - 0.264i)T \)
67 \( 1 + (0.166 - 0.986i)T \)
71 \( 1 + (-0.824 + 0.565i)T \)
73 \( 1 + (-0.460 + 0.887i)T \)
79 \( 1 + (-0.920 + 0.390i)T \)
83 \( 1 + (-0.210 - 0.977i)T \)
89 \( 1 + (-0.122 - 0.992i)T \)
97 \( 1 + (-0.911 + 0.410i)T \)
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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

−25.40744242351596165465528080433, −24.81270124883036448863825521048, −24.09098316837330901821685942776, −22.64258187248490116541176473643, −22.206132313610256434752401197063, −20.91343434370848911811272560110, −20.48503285728217151088769172851, −18.93744206127298890819380972755, −18.13956341103937405567753289180, −17.17370747761576280122736514497, −16.29361239920811459705491276350, −15.11895721703195505004314470166, −14.45254089952390036837932285534, −13.58250505522777513195140205704, −13.30463902194444085217595047921, −11.39573581409662076550236054198, −10.13899890250614684146464047073, −9.21922705785090685825023252891, −8.29423622952075916520796274108, −7.26244079573946024917090868284, −6.39506112198950223096325498005, −5.08218695472783272645154931174, −3.92027596263984812521571344710, −3.000790210758635092862706825798, −1.26894265875887211582886996127, 1.461670311354387903006385627258, 1.82805547822752898236942247200, 3.1792080458914964075522778955, 4.36798785938786498446087722893, 5.45365309560408080151316872174, 6.73744678010207332844368016890, 8.60368023534194669088976910575, 8.9714103764600922213361859908, 9.796013689512541957327235018923, 11.194143169911951891982034755877, 12.1943084183553294249703984985, 13.098472891625919778416144627, 13.83898033115155207031569968149, 14.6408913365753852265938024693, 15.576018538050777029146924005897, 17.302642322771420213366149751, 18.10920495185294161333930665072, 18.900531329616751158576071871327, 19.977554969863778847737383404979, 20.54679740160557620367615992667, 21.57464683197023396733138283082, 21.851711587207798209038529325655, 23.39426360745740226974012536422, 24.44154789569304991621660245489, 24.97389754660109610364500702337

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