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Properties

Label 1-1045-1045.189-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.0219 + 0.999i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s + 21-s − 23-s + (0.809 + 0.587i)24-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s + 21-s − 23-s + (0.809 + 0.587i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.0219 + 0.999i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.0219 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.758066028 + 1.719832472i\)
\(L(\frac12)\) \(\approx\) \(1.758066028 + 1.719832472i\)
\(L(1)\) \(\approx\) \(1.629514152 + 0.6145153309i\)
\(L(1)\) \(\approx\) \(1.629514152 + 0.6145153309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−21.183647824923817033779170473126, −20.700610911737716766348731074718, −20.01962463885655597000524105671, −19.53244802751086340491574025390, −18.29911348820409810824319538820, −17.44135030534633101442622863299, −16.333515813303329204140793321686, −15.70646525710010127992495083957, −15.02810981313017565889037045621, −13.992004642671172835342167819678, −13.72941288638916370910754072371, −12.79280737305743668250719559967, −11.476757777844845053238266620101, −11.109521943891513868498696051318, −10.20803529648277056158540909510, −9.684095477996386263350512564723, −8.542804725397515278744260677932, −7.584658580766517080379044654680, −6.33121846304769382243790523795, −5.52206545044516520456675116095, −4.37821695324002447874244229085, −4.11811204978946636973566824357, −3.0604123167326531945428755504, −2.15453409042674043502035412178, −0.73017361818717432113446046031, 1.61036198836225163651328647215, 2.401738086255448559506162587656, 3.36674553449239030678131716305, 4.41747545748980578920344753649, 5.50760207971898920559518940119, 6.32011558021839988908850446920, 6.79544889630490647601635609885, 8.07250123276875296713046323226, 8.42967739410292905320145081636, 9.2858002345733657618216001133, 11.00578554845193018925803503658, 11.667678791477422776129531541238, 12.48879956758018076913166487669, 12.9855306423227929744503396662, 14.129555613207375350235554753951, 14.27765387253752607822692027724, 15.51219645817455272256740864449, 15.90058213982295355323801375917, 17.16492908340243969426842308586, 17.80660288004752909779486944980, 18.496539520906940102647418513018, 19.37116469871119597733434840862, 20.31245660336784542343775679913, 21.05425690868370574967548733104, 21.86523547496070693570958213708

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