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Properties

Label 1-2019-2019.2018-r1-0-0
Degree $1$
Conductor $2019$
Sign $1$
Analytic cond. $216.971$
Root an. cond. $216.971$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 17-s − 19-s + 20-s − 22-s − 23-s + 25-s − 26-s + 28-s − 29-s − 31-s − 32-s − 34-s + 35-s + 37-s + 38-s − 40-s + 41-s + ⋯
L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 17-s − 19-s + 20-s − 22-s − 23-s + 25-s − 26-s + 28-s − 29-s − 31-s − 32-s − 34-s + 35-s + 37-s + 38-s − 40-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2019 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2019 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2019\)    =    \(3 \cdot 673\)
Sign: $1$
Analytic conductor: \(216.971\)
Root analytic conductor: \(216.971\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2019} (2018, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2019,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.581759190\)
\(L(\frac12)\) \(\approx\) \(2.581759190\)
\(L(1)\) \(\approx\) \(1.118669238\)
\(L(1)\) \(\approx\) \(1.118669238\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
673 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−19.8015645915458631417236615256, −18.66812487480450238251198694440, −18.34175966316775643777543217581, −17.61131128761256305472681788746, −16.86969657168099027968402922196, −16.56636221992063874010865368155, −15.41616750993446863324646579184, −14.522791865907273324694216115757, −14.23219378342894819785299418909, −13.06149102939022830462856885835, −12.22561214969561943352547835306, −11.31685938720214589137189596794, −10.843756846461579454460904386541, −9.98332896456712124495774408030, −9.25883937032673340690216918466, −8.6196149863422329718514240429, −7.88845807713250847623716069622, −7.004695145879791837463799600382, −5.991303701433148570803290356358, −5.73581308263650638240690533043, −4.333497275689258812742246731256, −3.36152232368201662050587027745, −2.07498195617091064987124074658, −1.636642357316434004182181599147, −0.80118824758378985412761499937, 0.80118824758378985412761499937, 1.636642357316434004182181599147, 2.07498195617091064987124074658, 3.36152232368201662050587027745, 4.333497275689258812742246731256, 5.73581308263650638240690533043, 5.991303701433148570803290356358, 7.004695145879791837463799600382, 7.88845807713250847623716069622, 8.6196149863422329718514240429, 9.25883937032673340690216918466, 9.98332896456712124495774408030, 10.843756846461579454460904386541, 11.31685938720214589137189596794, 12.22561214969561943352547835306, 13.06149102939022830462856885835, 14.23219378342894819785299418909, 14.522791865907273324694216115757, 15.41616750993446863324646579184, 16.56636221992063874010865368155, 16.86969657168099027968402922196, 17.61131128761256305472681788746, 18.34175966316775643777543217581, 18.66812487480450238251198694440, 19.8015645915458631417236615256

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