Properties

Degree 1
Conductor 19
Sign $0.0540 + 0.998i$
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0540 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0540 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(19\)
\( \varepsilon \)  =  $0.0540 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{19} (14, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 19,\ (1:\ ),\ 0.0540 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8336681241 + 0.7897236010i$
$L(\frac12,\chi)$  $\approx$  $0.8336681241 + 0.7897236010i$
$L(\chi,1)$  $\approx$  0.8659980343 + 0.4996074426i
$L(1,\chi)$  $\approx$  0.8659980343 + 0.4996074426i

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

−39.63374680849103885239582166312, −38.49455254722078840233510547241, −37.00350200897032593883902485315, −36.16322817316186247903413181750, −35.43977808784044782793338781848, −33.109418337386049025864922728658, −31.63413446829028814577362645968, −30.37543232456466401012598387362, −29.12701165878448250281289003919, −27.80678626863022828677426043446, −26.1506096505369850850793181241, −25.38822484094255058771609581434, −23.58114883222355613404709556085, −21.09085499144184530941001194642, −20.3010383606031585671003078028, −19.14907221039707331665919108201, −17.511353504250384558259542042165, −15.98594260188944482908090875857, −13.51126313642133614801819295360, −12.493432718088299094077896643818, −10.13112479843628132657938834286, −8.82738218161068529665525770797, −7.38330721431896536837684474762, −3.765895400747908765216110255704, −1.46008805795875957859250184976, 2.80964420013225637144552555427, 6.00945940147099369450667486785, 7.90200318341973777138315653818, 9.34296826203185857800902821557, 10.75058408725432273707517000394, 13.751392698154821500436397584139, 15.093582638731824697435301400178, 16.12950676783959421902281807179, 18.46291661418661357390712291166, 19.042380304046326092090797405857, 20.9422991982478007054541632513, 22.67979873610811110893275123939, 24.69528315306723419739509681820, 25.78470197190333292775941621848, 26.58973966498685181395632999275, 28.00826607146601583770636644008, 29.736961082421553137842704573930, 31.49126067067186211761281410492, 32.68893835135116473265447986895, 34.075869513997433776005364223275, 35.22391808574180821578932444907, 36.79375119221932129368465258291, 37.78862132114505789486989335660, 38.53985114461569917384917384149, 41.20793173817761339204221149711

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