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Properties

Label 1-1045-1045.129-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.486 - 0.873i$
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  + (−0.848 + 0.529i)2-s + (0.559 − 0.829i)3-s + (0.438 − 0.898i)4-s + (−0.0348 + 0.999i)6-s + (−0.104 + 0.994i)7-s + (0.104 + 0.994i)8-s + (−0.374 − 0.927i)9-s + (−0.5 − 0.866i)12-s + (0.615 − 0.788i)13-s + (−0.438 − 0.898i)14-s + (−0.615 − 0.788i)16-s + (−0.374 + 0.927i)17-s + (0.809 + 0.587i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.882 + 0.469i)24-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.559 − 0.829i)3-s + (0.438 − 0.898i)4-s + (−0.0348 + 0.999i)6-s + (−0.104 + 0.994i)7-s + (0.104 + 0.994i)8-s + (−0.374 − 0.927i)9-s + (−0.5 − 0.866i)12-s + (0.615 − 0.788i)13-s + (−0.438 − 0.898i)14-s + (−0.615 − 0.788i)16-s + (−0.374 + 0.927i)17-s + (0.809 + 0.587i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.882 + 0.469i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.486 - 0.873i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.486 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9102632720 - 0.5352751883i\)
\(L(\frac12)\) \(\approx\) \(0.9102632720 - 0.5352751883i\)
\(L(1)\) \(\approx\) \(0.8412095071 - 0.1146407290i\)
\(L(1)\) \(\approx\) \(0.8412095071 - 0.1146407290i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
3 \( 1 + (0.559 - 0.829i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.615 - 0.788i)T \)
17 \( 1 + (-0.374 + 0.927i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.997 + 0.0697i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.559 - 0.829i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.241 - 0.970i)T \)
53 \( 1 + (0.990 + 0.139i)T \)
59 \( 1 + (0.241 + 0.970i)T \)
61 \( 1 + (0.882 - 0.469i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.990 + 0.139i)T \)
73 \( 1 + (0.961 + 0.275i)T \)
79 \( 1 + (0.0348 + 0.999i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.848 - 0.529i)T \)
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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

−21.27469315073590587512543520113, −20.813793180847433741959412386612, −20.14322405676949744533950210181, −19.456165077636135972040785161967, −18.78880602858268519379903494374, −17.704426452232606705043526108205, −17.02160281258595345544154922506, −16.12621311186226406044583542159, −15.84559770658397881199715918248, −14.59229269298693732826978924691, −13.64298676990224521254123024719, −13.184495808347617981569262633404, −11.685797809620136563919118181072, −11.190197994168361667262454503972, −10.34929256202464817707744008168, −9.59887756354999193791685072068, −9.06008809644323132536663597885, −8.0636652573744594636221721014, −7.36455453831817371487718047339, −6.395754108328794820353331564731, −4.84463124856798251445842564811, −3.9733962692577628347536519238, −3.31913490658865155673716042571, −2.26780190894581986216950800862, −1.137838563576915315172922240319, 0.60556669933602601170184188512, 1.895817940097037726027532772150, 2.49975790739758711859030439740, 3.75840672257297726690151939324, 5.42236837844512792834709350211, 6.0434350906062452103782558012, 6.82528850732431868357879914002, 7.771603091766127688348813259693, 8.64670934754062314825418252007, 8.82231892952123514711549044213, 10.030541442268713052533756004203, 10.89966131484528346441500194225, 11.93680621278002702456435269542, 12.692064496953408851632354688577, 13.59114528909514190871011625606, 14.55031186215483259409619585280, 15.23476505728598439986318672681, 15.7536454876267473197232138981, 16.93165681281184033347625701185, 17.696190986475033091888666126672, 18.359860199837849795422744485603, 18.915054006523688149849527456764, 19.588523249877605568916243962167, 20.41190043306999757888156822760, 21.09005585970271391059358802714

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