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Properties

Label 1-19-19.10-r1-0-0
Degree $1$
Conductor $19$
Sign $0.980 + 0.196i$
Analytic cond. $2.04183$
Root an. cond. $2.04183$
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  + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.939 + 0.342i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (−0.766 − 0.642i)14-s + (−0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.939 + 0.342i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (−0.766 − 0.642i)14-s + (−0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $0.980 + 0.196i$
Analytic conductor: \(2.04183\)
Root analytic conductor: \(2.04183\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (1:\ ),\ 0.980 + 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.973817970 + 0.1957502106i\)
\(L(\frac12)\) \(\approx\) \(1.973817970 + 0.1957502106i\)
\(L(1)\) \(\approx\) \(1.660301634 + 0.1000769021i\)
\(L(1)\) \(\approx\) \(1.660301634 + 0.1000769021i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−40.59294116672808166460496738484, −39.43514244239238140166406361752, −37.59559707179108460393255301060, −35.973527485234823231916095648758, −34.803493837508893892373719786704, −33.56735239520936400902089836715, −31.971763523267286440591611540525, −31.08745948920282231489892560645, −29.30676061222835925079542104486, −28.82642615782907725515631340688, −25.91580962283144924963551746847, −24.763826382247811897420968436766, −23.9715252118784483286726638424, −22.31425436691089341533025009498, −21.071723502346221857968274427526, −19.18057587678369335750470180227, −17.40193068958950159836810892492, −15.98467714326647205240586893365, −13.88679509946100288128298854289, −12.95799316677742842970647875788, −11.635351450120620218563245761694, −8.67354534726236485864325823859, −6.59142772587323881424794900382, −5.37011900372183134500816131601, −2.39276444415460360932038984285, 3.007290648223576261709094943072, 4.844020882450831383706659292615, 6.602885669054097391483674015466, 10.02191643776843284865225332735, 10.74952018439617717345715267411, 12.96375432162960380202140802186, 14.46497014729291701213093981024, 15.707003782804773143765595485240, 17.4947600170617978376102149037, 19.887679113267158044339210294658, 21.03384680967809964075526831563, 22.36670876315206745340868276411, 23.121524521330731448722189255957, 25.30351526184014801392532936517, 26.60139665133819495386195027845, 28.47610320248521675237576170891, 29.54255930682118577613500707697, 30.95069409726045263827618158922, 32.665089019440368273397279488361, 33.15017263515050345128282507175, 34.39380163641819383680604948446, 36.87537487458207336883900507342, 38.03673137821961937974907766312, 39.0901803465859986416684212495, 40.0739957719464559867467688341

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