Properties

Degree 1
Conductor 19
Sign $0.672 + 0.740i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(19\)
\( \varepsilon \)  =  $0.672 + 0.740i$
motivic weight  =  \(0\)
character  :  $\chi_{19} (4, \cdot )$
Sato-Tate  :  $\mu(9)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 19,\ (0:\ ),\ 0.672 + 0.740i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6544410037 + 0.2897829633i$
$L(\frac12,\chi)$  $\approx$  $0.6544410037 + 0.2897829633i$
$L(\chi,1)$  $\approx$  0.9346598879 + 0.3179639679i
$L(1,\chi)$  $\approx$  0.9346598879 + 0.3179639679i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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