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Properties

Label 1-1045-1045.984-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.382 - 0.924i$
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  + (−0.997 − 0.0697i)2-s + (−0.615 − 0.788i)3-s + (0.990 + 0.139i)4-s + (0.559 + 0.829i)6-s + (0.978 − 0.207i)7-s + (−0.978 − 0.207i)8-s + (−0.241 + 0.970i)9-s + (−0.5 − 0.866i)12-s + (0.961 − 0.275i)13-s + (−0.990 + 0.139i)14-s + (0.961 + 0.275i)16-s + (0.241 + 0.970i)17-s + (0.309 − 0.951i)18-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.438 + 0.898i)24-s + ⋯
L(s)  = 1  + (−0.997 − 0.0697i)2-s + (−0.615 − 0.788i)3-s + (0.990 + 0.139i)4-s + (0.559 + 0.829i)6-s + (0.978 − 0.207i)7-s + (−0.978 − 0.207i)8-s + (−0.241 + 0.970i)9-s + (−0.5 − 0.866i)12-s + (0.961 − 0.275i)13-s + (−0.990 + 0.139i)14-s + (0.961 + 0.275i)16-s + (0.241 + 0.970i)17-s + (0.309 − 0.951i)18-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.438 + 0.898i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.382 - 0.924i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (984, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.382 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.019912061 - 0.6817492799i\)
\(L(\frac12)\) \(\approx\) \(1.019912061 - 0.6817492799i\)
\(L(1)\) \(\approx\) \(0.6750902193 - 0.2032604294i\)
\(L(1)\) \(\approx\) \(0.6750902193 - 0.2032604294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.997 - 0.0697i)T \)
3 \( 1 + (-0.615 - 0.788i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (0.961 - 0.275i)T \)
17 \( 1 + (0.241 + 0.970i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.374 + 0.927i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.848 - 0.529i)T \)
53 \( 1 + (-0.719 - 0.694i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (0.438 - 0.898i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.719 - 0.694i)T \)
73 \( 1 + (-0.0348 - 0.999i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.997 - 0.0697i)T \)
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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

−21.27553745288006350960647631683, −20.72117443743199117931795028369, −20.12504236426937054003626141205, −18.87884178600771416726420122593, −18.24576385938843222013608913319, −17.55325156075583575999443213519, −16.96430129013895227500092922958, −15.954229150203893428508748323497, −15.63578294577316550688886925156, −14.68041883413039152107683518002, −13.839899329370041693435744759888, −12.3405181479100467818558783473, −11.43952091151502375974748155732, −11.23934565742869180254964311331, −10.2481040269379828118534727728, −9.44102180369712063532061075201, −8.74439012384363214575558199444, −7.8714786412061477083773501826, −6.894548290942277821436853680351, −5.88458890302907954634288659043, −5.224412569786291444169200522159, −4.08645186281387805293547068591, −2.99172787446956287611577758448, −1.69234512347219729592359443250, −0.74436559347486527248767493071, 0.57184006693362695777074932576, 1.44021328989037113723788869230, 2.132664862831404232099453640989, 3.48526057808662298680493353368, 4.85329507721468070141758862920, 5.95398240849614738597481271442, 6.53732135637916901697449967420, 7.64102850442178063878752866195, 8.11943629508584436388808363501, 8.8755856401281611508126408548, 10.195931187426236185641226734572, 11.00048782860262743355355882497, 11.266335685724802336756028440566, 12.416657961278351828173977513578, 12.95393271439252501928800642602, 14.21994613423864218578240117235, 14.932604551428321492309065256730, 16.144915302689133016008858688294, 16.67148958715854014616944976013, 17.513934723546560181462922648561, 18.16720785789781524947348683432, 18.50765764764877429794965488175, 19.58238592181514787244495451890, 20.1699733086882665177512463386, 21.12143187875981185103250519629

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