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Properties

Label 1-19-19.5-r0-0-0
Degree $1$
Conductor $19$
Sign $0.672 - 0.740i$
Analytic cond. $0.0882356$
Root an. cond. $0.0882356$
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  + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (−0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.939 + 0.342i)13-s + (0.173 + 0.984i)14-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (−0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.939 + 0.342i)13-s + (0.173 + 0.984i)14-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $0.672 - 0.740i$
Analytic conductor: \(0.0882356\)
Root analytic conductor: \(0.0882356\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (0:\ ),\ 0.672 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6544410037 - 0.2897829633i\)
\(L(\frac12)\) \(\approx\) \(0.6544410037 - 0.2897829633i\)
\(L(1)\) \(\approx\) \(0.9346598879 - 0.3179639679i\)
\(L(1)\) \(\approx\) \(0.9346598879 - 0.3179639679i\)

Euler product

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

−40.73511331641937823479158654581, −39.436729318278928558130480873611, −39.05853107196549317818200168902, −36.360917937869011130585856026737, −35.244302891362101235177175999080, −33.83264566228813203487427134439, −32.8316796953852043239796802795, −31.90409335249010647702215075702, −29.94940283780059796569500959358, −28.75773884990395915081671481112, −27.13016087682013018253057362276, −25.518978292967395922499891031447, −23.90341813148348010068305725289, −23.08229152770435882453451772191, −21.63487538719064430662036135996, −20.31672061163687114035918764775, −17.41084230085729508056606187233, −16.690837499994326190943952264138, −15.313347240163891215823812917713, −13.18528617998266692040660218922, −12.13978869658205557460065589147, −9.93853825544030189827511416730, −7.42932535066366940295882182109, −5.58954103716691336093904128356, −4.27062371721823059571900449662, 2.733275043504043661147892199151, 5.36365180418947272260918420061, 6.689133903682895395393297028686, 10.06672385430339217931313091097, 11.41801889446997423804826287906, 12.659780578684548636652178861829, 14.34529306297111655595173658600, 16.06175781077484605233344018615, 18.34799293217231120575904428118, 19.133396743653995523374837092712, 21.56008653367801169765762177996, 22.27726142506495215789275244009, 23.5008957159694514779408412246, 24.9571305143292949799141483045, 27.16850432518058979780876797894, 28.84103953152572040482629847577, 29.47377580155829746091329365937, 30.830174886728508200911079083481, 32.248849244562352323535497177277, 33.93925700259191197563304657716, 34.6382753336706649716939919179, 36.71974276195177520349265877379, 38.244943711593109179644650720196, 39.08536957417750106224680368853, 40.55589522570808690351019129377

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