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Properties

Label 1-19-19.9-r0-0-0
Degree $1$
Conductor $19$
Sign $0.944 - 0.327i$
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  + (−0.939 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(0.0882356\)
Root analytic conductor: \(0.0882356\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (0:\ ),\ 0.944 - 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4813597783 - 0.08095972326i\)
\(L(\frac12)\) \(\approx\) \(0.4813597783 - 0.08095972326i\)
\(L(1)\) \(\approx\) \(0.6806203626 - 0.07154573942i\)
\(L(1)\) \(\approx\) \(0.6806203626 - 0.07154573942i\)

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

−40.19556239541361684489349653984, −39.142587598501784622929911836169, −37.65358685340258880853658006662, −37.17482093185859369146813608588, −35.48185200121351473056659299455, −34.13473326392034049888800239490, −32.72455619776752419519397492763, −31.431043544026281731752857302092, −29.302164803892645773900440983000, −28.40462223690512318423029732428, −27.210626350268492535647292689404, −25.8074760332197126106624974940, −24.84055154790605485653744528511, −22.04243097401595638335122920943, −21.09369041967365418744910878486, −19.84018281515610898761976834369, −18.07215790170884784327140245890, −16.56452895686258233194656461370, −15.49926570895319878901602633044, −13.02144713119484306788497190437, −11.00709089461077718283415647784, −9.52751247063341249444310158906, −8.568989446161238201271795823444, −5.656452297566206800854634914471, −2.87454712437612373765181195590, 2.13818063440820276534060956054, 6.384715706532220203928471261360, 7.35639597684667641097708975505, 9.37779373656853152674600006352, 10.971758619635417447967778029, 13.24067770016994068392736936232, 14.73284120125166592095395922477, 16.88801311771858308770908183410, 18.01493004225494195693449722914, 19.14569065365657881342526990595, 20.53443112178814757544021108462, 22.97348373174334988688227698800, 24.32774447659544996710004942010, 25.8102237891815163877704406034, 26.35170356332628860499363672334, 28.655702383358154296641632840894, 29.435748644174708771084203921301, 30.7532468321009262631444666575, 33.01215810722231408734321347350, 34.02705925837258318591857403739, 35.53975188720620221090336426655, 36.39723933268938600167310368002, 37.43253926972937848484461994432, 38.91217747713898601121875846822, 41.03446028747668469681275718764

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