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Properties

Label 1-1045-1045.272-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.227 + 0.973i$
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  + (−0.469 + 0.882i)2-s + (−0.0697 + 0.997i)3-s + (−0.559 − 0.829i)4-s + (−0.848 − 0.529i)6-s + (0.994 + 0.104i)7-s + (0.994 − 0.104i)8-s + (−0.990 − 0.139i)9-s + (0.866 − 0.5i)12-s + (0.927 − 0.374i)13-s + (−0.559 + 0.829i)14-s + (−0.374 + 0.927i)16-s + (−0.139 − 0.990i)17-s + (0.587 − 0.809i)18-s + (−0.173 + 0.984i)21-s + (−0.342 + 0.939i)23-s + (0.0348 + 0.999i)24-s + ⋯
L(s)  = 1  + (−0.469 + 0.882i)2-s + (−0.0697 + 0.997i)3-s + (−0.559 − 0.829i)4-s + (−0.848 − 0.529i)6-s + (0.994 + 0.104i)7-s + (0.994 − 0.104i)8-s + (−0.990 − 0.139i)9-s + (0.866 − 0.5i)12-s + (0.927 − 0.374i)13-s + (−0.559 + 0.829i)14-s + (−0.374 + 0.927i)16-s + (−0.139 − 0.990i)17-s + (0.587 − 0.809i)18-s + (−0.173 + 0.984i)21-s + (−0.342 + 0.939i)23-s + (0.0348 + 0.999i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.227 + 0.973i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.227 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9469607780 + 0.7509994101i\)
\(L(\frac12)\) \(\approx\) \(0.9469607780 + 0.7509994101i\)
\(L(1)\) \(\approx\) \(0.7642470997 + 0.4939905114i\)
\(L(1)\) \(\approx\) \(0.7642470997 + 0.4939905114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.469 + 0.882i)T \)
3 \( 1 + (-0.0697 + 0.997i)T \)
7 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (0.927 - 0.374i)T \)
17 \( 1 + (-0.139 - 0.990i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.438 - 0.898i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.997 + 0.0697i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.694 - 0.719i)T \)
53 \( 1 + (0.788 + 0.615i)T \)
59 \( 1 + (0.719 - 0.694i)T \)
61 \( 1 + (-0.0348 + 0.999i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (-0.615 - 0.788i)T \)
73 \( 1 + (0.970 - 0.241i)T \)
79 \( 1 + (0.848 - 0.529i)T \)
83 \( 1 + (0.743 + 0.669i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.469 - 0.882i)T \)
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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

−21.21444977178343164381545735548, −20.43457116413144388717327378004, −19.8233941774857722175835470752, −18.96324603237283668803986131928, −18.23438094571453741237123370487, −17.86540106834737109558561361626, −16.97141300053880512888099928600, −16.28259393054717939956920112844, −14.74465217417704302319490052895, −14.107542707065075948416985808654, −13.227101347566055384993170054830, −12.59835863710340555724232834520, −11.75051825416743502538760010725, −11.053780111347130232667190199376, −10.510962155879084689726844751515, −9.118197326193875903118338395526, −8.40318238044393692454858453689, −7.893331514775094512884583251710, −6.87659439597094549077196600700, −5.86463738141745197532329947892, −4.66940737352258467799273987782, −3.74202570409551634113929821448, −2.54802759490066564129587505725, −1.69522412518107718739398467194, −0.99978573413477875347120020326, 0.79915509594097483218113433735, 2.209395075139967336252187409546, 3.70854512744232654716713491415, 4.51325714420162754472513896429, 5.43508365991570608884103482626, 5.89944814841043224775502928374, 7.239909449716729875589332143132, 8.04281814494210178540083544156, 8.850441924241816937307730596073, 9.451934796757777212500531042573, 10.46985500095963654623983716175, 11.06939472679256246383271426076, 11.89534912378458194157245980356, 13.488394352050139665469217414361, 14.033996679273996686721313897267, 14.92350072770729955429451323643, 15.49238349292277339526926932363, 16.179798251780673757906983299, 16.89475862924475048024271095765, 17.970884974558805737512937878076, 18.02256896003511414678408588981, 19.359517243963526077141839597753, 20.22866460482918932083444999498, 20.90366679146619702417313883676, 21.7431804874706943785493085969

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