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Properties

Label 1-2019-2019.119-r0-0-0
Degree $1$
Conductor $2019$
Sign $0.148 - 0.988i$
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  + (−0.793 − 0.608i)2-s + (0.258 + 0.965i)4-s + (0.673 + 0.739i)5-s + (−0.666 + 0.745i)7-s + (0.382 − 0.923i)8-s + (−0.0840 − 0.996i)10-s + (−0.00934 − 0.999i)11-s + (0.294 + 0.955i)13-s + (0.982 − 0.185i)14-s + (−0.866 + 0.5i)16-s + (0.879 − 0.475i)17-s + (0.391 − 0.920i)19-s + (−0.539 + 0.841i)20-s + (−0.601 + 0.799i)22-s + (−0.900 − 0.433i)23-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)2-s + (0.258 + 0.965i)4-s + (0.673 + 0.739i)5-s + (−0.666 + 0.745i)7-s + (0.382 − 0.923i)8-s + (−0.0840 − 0.996i)10-s + (−0.00934 − 0.999i)11-s + (0.294 + 0.955i)13-s + (0.982 − 0.185i)14-s + (−0.866 + 0.5i)16-s + (0.879 − 0.475i)17-s + (0.391 − 0.920i)19-s + (−0.539 + 0.841i)20-s + (−0.601 + 0.799i)22-s + (−0.900 − 0.433i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2019 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2019 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2019\)    =    \(3 \cdot 673\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(9.37619\)
Root analytic conductor: \(9.37619\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2019} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2019,\ (0:\ ),\ 0.148 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6878049771 - 0.5920112904i\)
\(L(\frac12)\) \(\approx\) \(0.6878049771 - 0.5920112904i\)
\(L(1)\) \(\approx\) \(0.7448736334 - 0.1469229694i\)
\(L(1)\) \(\approx\) \(0.7448736334 - 0.1469229694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
673 \( 1 \)
good2 \( 1 + (-0.793 - 0.608i)T \)
5 \( 1 + (0.673 + 0.739i)T \)
7 \( 1 + (-0.666 + 0.745i)T \)
11 \( 1 + (-0.00934 - 0.999i)T \)
13 \( 1 + (0.294 + 0.955i)T \)
17 \( 1 + (0.879 - 0.475i)T \)
19 \( 1 + (0.391 - 0.920i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.608 - 0.793i)T \)
31 \( 1 + (0.700 - 0.713i)T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + (0.999 + 0.0280i)T \)
43 \( 1 + (0.475 - 0.879i)T \)
47 \( 1 + (0.158 - 0.987i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (-0.442 - 0.896i)T \)
61 \( 1 + (-0.616 + 0.787i)T \)
67 \( 1 + (-0.713 + 0.700i)T \)
71 \( 1 + (-0.303 + 0.952i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (-0.356 - 0.934i)T \)
83 \( 1 + (0.856 - 0.516i)T \)
89 \( 1 + (0.999 + 0.0186i)T \)
97 \( 1 + (-0.593 + 0.804i)T \)
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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

−20.09646877289641512356787838579, −19.40149217897786006566812552301, −18.45440915922344166198998242209, −17.66325676141870470537913574481, −17.3240486121448621305033540013, −16.43449323736471025231424322624, −16.03583886674948493022708033012, −15.16030458526303092985070745609, −14.21813580080836965381331068995, −13.720066759053533583471773798374, −12.63793646002243404714738151686, −12.24877742159632220584615978959, −10.74093297792505032052916608285, −10.21397401366793354966370407569, −9.68574054650288858785039231568, −9.0012368318663440012942179418, −7.85805672008822884926599944718, −7.60477540046742525828659052217, −6.40792600288242437492706875053, −5.84201495677201741985840011934, −5.07844106181491976147309844193, −4.07820149819993522236360692587, −2.91949353347984745948901460561, −1.580953478360812382776458101692, −1.11765126519257546706453040377, 0.43925440368988589235472806413, 1.793052716870052621615025564851, 2.55446343303899465242301982947, 3.19152461952319088285991088891, 4.042842353454317457694299779215, 5.51733353353204164460635272296, 6.23519680856695126110452259385, 6.938050841830171524155203746173, 7.845772660193907985942343034, 8.86614729635445038877975297402, 9.337566238267813326945744114022, 10.03521653887283903694708826995, 10.79037086214336561529537076002, 11.6616169696102651123860215726, 12.021411982843138975902525810134, 13.276698572617944982565405511472, 13.65286796645507264272645566728, 14.57772636093562624370405063078, 15.711202966250132037486764286080, 16.20311429587714323685467700573, 16.97104898744524328124692073368, 17.79375278554438816064336542174, 18.50631578953888058533231311963, 19.03082499297227733815857661885, 19.338396398774085212530233997637

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