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Properties

Label 1-2019-2019.290-r0-0-0
Degree $1$
Conductor $2019$
Sign $-0.811 - 0.585i$
Analytic cond. $9.37619$
Root an. cond. $9.37619$
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  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (0.00934 − 0.999i)5-s + (0.167 + 0.985i)7-s + (0.382 − 0.923i)8-s + (0.990 + 0.139i)10-s + (0.158 + 0.987i)11-s + (−0.930 + 0.365i)13-s + (−0.999 + 0.0373i)14-s + (0.866 + 0.5i)16-s + (0.539 + 0.841i)17-s + (−0.524 + 0.851i)19-s + (−0.267 + 0.963i)20-s + (−0.999 + 0.0280i)22-s + (−0.222 − 0.974i)23-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (0.00934 − 0.999i)5-s + (0.167 + 0.985i)7-s + (0.382 − 0.923i)8-s + (0.990 + 0.139i)10-s + (0.158 + 0.987i)11-s + (−0.930 + 0.365i)13-s + (−0.999 + 0.0373i)14-s + (0.866 + 0.5i)16-s + (0.539 + 0.841i)17-s + (−0.524 + 0.851i)19-s + (−0.267 + 0.963i)20-s + (−0.999 + 0.0280i)22-s + (−0.222 − 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2019 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2019 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2019\)    =    \(3 \cdot 673\)
Sign: $-0.811 - 0.585i$
Analytic conductor: \(9.37619\)
Root analytic conductor: \(9.37619\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2019} (290, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2019,\ (0:\ ),\ -0.811 - 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1499312810 + 0.4641275013i\)
\(L(\frac12)\) \(\approx\) \(-0.1499312810 + 0.4641275013i\)
\(L(1)\) \(\approx\) \(0.6355606743 + 0.4140373340i\)
\(L(1)\) \(\approx\) \(0.6355606743 + 0.4140373340i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
673 \( 1 \)
good2 \( 1 + (-0.130 + 0.991i)T \)
5 \( 1 + (0.00934 - 0.999i)T \)
7 \( 1 + (0.167 + 0.985i)T \)
11 \( 1 + (0.158 + 0.987i)T \)
13 \( 1 + (-0.930 + 0.365i)T \)
17 \( 1 + (0.539 + 0.841i)T \)
19 \( 1 + (-0.524 + 0.851i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.991 - 0.130i)T \)
31 \( 1 + (-0.586 + 0.810i)T \)
37 \( 1 + (0.781 + 0.623i)T \)
41 \( 1 + (-0.888 - 0.458i)T \)
43 \( 1 + (-0.841 - 0.539i)T \)
47 \( 1 + (-0.425 - 0.904i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (0.997 + 0.0654i)T \)
61 \( 1 + (-0.958 - 0.285i)T \)
67 \( 1 + (0.810 - 0.586i)T \)
71 \( 1 + (-0.861 - 0.508i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (-0.0840 + 0.996i)T \)
83 \( 1 + (-0.978 - 0.204i)T \)
89 \( 1 + (0.949 + 0.312i)T \)
97 \( 1 + (0.982 - 0.185i)T \)
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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

−19.643997513520100262570489525696, −18.906335453752886570624529758266, −18.12723457448020585573521577028, −17.5125488495809668131683940521, −16.856945605366561996824477755571, −15.94873396486093235651649288749, −14.68527874656420404500964965568, −14.352892097281170305171735125009, −13.47526742641776098389811863006, −13.04675459316594047761527719834, −11.633709201715705126915968184498, −11.49013661641475322914953563498, −10.58802192694897143968128567305, −10.00399979313821974332590662053, −9.35810387431706730453506821480, −8.19514776875012608666122544297, −7.554954410383285054867418179107, −6.7814008344303012687178337708, −5.63785362540892579805356854855, −4.732870872425998511365917938909, −3.82035748659824356918248423261, −3.08763315820619845194514363802, −2.46343309370414918715701142513, −1.23563760790587683697041120271, −0.18603127373622536830829379618, 1.40902520242943747157622517118, 2.18670458815266342172724916271, 3.72488731961131795631559550031, 4.66735602993194358897578507092, 5.06887824845951667793066845798, 5.97958853497053019599841144173, 6.70507485450430585961031388977, 7.722029152660248584312681645938, 8.4372449454552358460338494376, 8.8893192536353190707131507560, 9.851476684308829743299027906486, 10.28664242598659168174989792390, 12.02710979067842150991173142040, 12.282467948161796974745871265975, 12.93459855720434610327660876285, 13.99472442565804239415822375347, 14.84001175865667156098908019668, 15.113699307176242783638108517857, 16.09761273425952793171984415279, 16.79316054816487854288103965900, 17.20620347320577084640562495477, 18.09165746283687833486399709622, 18.72840645100948994322714751163, 19.55857610804419072504125052678, 20.236696770662862523254521568191

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