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Properties

Label 1-19-19.3-r1-0-0
Degree $1$
Conductor $19$
Sign $-0.766 + 0.642i$
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  + (−0.173 − 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.173 + 0.984i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.766 − 0.642i)13-s + (0.939 + 0.342i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.173 + 0.984i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.766 − 0.642i)13-s + (0.939 + 0.342i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(2.04183\)
Root analytic conductor: \(2.04183\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (1:\ ),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006540279335 + 0.01798753263i\)
\(L(\frac12)\) \(\approx\) \(0.006540279335 + 0.01798753263i\)
\(L(1)\) \(\approx\) \(0.4003261943 - 0.1023120427i\)
\(L(1)\) \(\approx\) \(0.4003261943 - 0.1023120427i\)

Euler product

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

−39.933443954540430756987766596570, −38.7319576241133433443562851343, −36.27500694395829898794590394093, −35.78513866670108032028068658837, −34.35490032841904331292881028432, −33.60174962412203491458045024734, −31.81661658709014940587772017330, −30.43241238919920332103315488844, −28.74242685164081015301764918591, −27.31345094494983871564259539018, −26.09355589856479562586144829588, −24.373054689141404202145379001565, −23.25949236454995257704793565974, −22.57185973340890259115544397021, −19.61454129002855771309572217025, −18.3605308153124098219189668275, −16.94315331681772467787605951502, −15.75307297528780652149410600, −13.89960291282131331636248843366, −12.208965794458144063116929434818, −10.19493963410200532885344100164, −7.6381323538639373403363326575, −6.80199131141896746241442476871, −4.62408465986036793558371161279, −0.01895639908022614299416416142, 3.49447125473922415012953094194, 5.3308864285875200315174416326, 8.475764952021204639766068265616, 10.16954559070287444750721866614, 11.6434871873614534188282603618, 12.66685849021041629323708921688, 15.3154386273385321159241779328, 16.77209204007393558656625257875, 18.49905319570608800860032805677, 19.85810375858308120501069516815, 21.46365988923275335619089858791, 22.449262516754894958551388501363, 23.854216770162870885673608038, 26.394081332691340773063062208137, 27.61824023406910104298345702641, 28.423462435887126999187323519968, 29.6946548093245793861945047195, 31.57600074723917889205151966813, 32.26416255317538221304288041299, 34.59737233421130858318266847354, 35.287741884806483203814877021280, 37.143997211393806916543385637632, 38.48224263709045878811378207483, 39.31340468270123629908990859772, 40.21566114557832722893207318034

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