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Properties

Label 1-1045-1045.288-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.122 + 0.992i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (−0.898 − 0.438i)2-s + (−0.927 + 0.374i)3-s + (0.615 + 0.788i)4-s + (0.997 + 0.0697i)6-s + (0.207 − 0.978i)7-s + (−0.207 − 0.978i)8-s + (0.719 − 0.694i)9-s + (−0.866 − 0.5i)12-s + (0.970 − 0.241i)13-s + (−0.615 + 0.788i)14-s + (−0.241 + 0.970i)16-s + (0.694 − 0.719i)17-s + (−0.951 + 0.309i)18-s + (0.173 + 0.984i)21-s + (−0.342 − 0.939i)23-s + (0.559 + 0.829i)24-s + ⋯
L(s)  = 1  + (−0.898 − 0.438i)2-s + (−0.927 + 0.374i)3-s + (0.615 + 0.788i)4-s + (0.997 + 0.0697i)6-s + (0.207 − 0.978i)7-s + (−0.207 − 0.978i)8-s + (0.719 − 0.694i)9-s + (−0.866 − 0.5i)12-s + (0.970 − 0.241i)13-s + (−0.615 + 0.788i)14-s + (−0.241 + 0.970i)16-s + (0.694 − 0.719i)17-s + (−0.951 + 0.309i)18-s + (0.173 + 0.984i)21-s + (−0.342 − 0.939i)23-s + (0.559 + 0.829i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.122 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1353894942 + 0.1531921699i\)
\(L(\frac12)\) \(\approx\) \(0.1353894942 + 0.1531921699i\)
\(L(1)\) \(\approx\) \(0.5014850642 - 0.1103253883i\)
\(L(1)\) \(\approx\) \(0.5014850642 - 0.1103253883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.898 - 0.438i)T \)
3 \( 1 + (-0.927 + 0.374i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (0.970 - 0.241i)T \)
17 \( 1 + (0.694 - 0.719i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (-0.990 + 0.139i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.374 - 0.927i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.469 + 0.882i)T \)
53 \( 1 + (0.275 - 0.961i)T \)
59 \( 1 + (-0.882 + 0.469i)T \)
61 \( 1 + (-0.559 + 0.829i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (-0.961 + 0.275i)T \)
73 \( 1 + (0.529 - 0.848i)T \)
79 \( 1 + (0.997 - 0.0697i)T \)
83 \( 1 + (-0.994 + 0.104i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.898 + 0.438i)T \)
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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

−21.25097500030493242383780546717, −20.18759134152330067965118669820, −19.16951758253961493913143237630, −18.524520353510591593588746055013, −18.19833831712984729655401966872, −17.18307247451861979121770068423, −16.68071193030451584225855130679, −15.735350720707837131313909330500, −15.250744409550241424253347396155, −14.17626190968686259967098335689, −13.11821601596287281496936687337, −12.13145092980197951380088908212, −11.45219952885825742594555962727, −10.80725281937590508487583393579, −9.86943527991877064123765695127, −8.98239251907503069644201315775, −8.120220137995502511544801333497, −7.366958820234981144814237018299, −6.32439171567139953458598233743, −5.75642185333127155379339488816, −5.1138636478926092042850973835, −3.58488064156243472502726566521, −1.962402160679396105869527137289, −1.47069551568663056958714820938, −0.0845548767864217040286477762, 0.83857751803282477677827416439, 1.66314959919463525308576987083, 3.280354709851527346915856334279, 3.9439412053060022929377172481, 5.04160072132797045472094862281, 6.1776327860497249689925696028, 7.01048659570100419680687644962, 7.7632848930128838506503003531, 8.81917232533374629273723191542, 9.73405030771699933340108541067, 10.50278301594071881417924458510, 10.962656383469264158591954815083, 11.75113453282406490018264050179, 12.59064792451320030136741441514, 13.43000726183304719548663487573, 14.59043554213416784732873087535, 15.71214767270186073524748902782, 16.42974363357380228272847591820, 16.80138045651755881931876382921, 17.78841905656564852212530924114, 18.24571245939040008448564160139, 19.053120924858364528440009108364, 20.21289736176851282000173319179, 20.70630881868737936014702227930, 21.27660865323161992557618532606

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