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Properties

Label 1-19-19.15-r1-0-0
Degree $1$
Conductor $19$
Sign $0.0540 - 0.998i$
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  + (−0.766 − 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

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

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $0.0540 - 0.998i$
Analytic conductor: \(2.04183\)
Root analytic conductor: \(2.04183\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (1:\ ),\ 0.0540 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8336681241 - 0.7897236010i\)
\(L(\frac12)\) \(\approx\) \(0.8336681241 - 0.7897236010i\)
\(L(1)\) \(\approx\) \(0.8659980343 - 0.4996074426i\)
\(L(1)\) \(\approx\) \(0.8659980343 - 0.4996074426i\)

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

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

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