# Properties

 Degree $1$ Conductor $2019$ Sign $1$ Motivic weight $0$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

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

## Invariants

 Degree: $$1$$ Conductor: $$2019$$    =    $$3 \cdot 673$$ Sign: $1$ Motivic weight: $$0$$ Character: $\chi_{2019} (2018, \cdot )$ Sato-Tate group: $\mu(2)$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 2019,\ (1:\ ),\ 1)$$

## Particular Values

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

## Euler product

$$L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}$$
$$L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}$$

## 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