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Properties

Label 1-1045-1045.754-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.626 + 0.779i$
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  + (−0.438 − 0.898i)2-s + (−0.374 + 0.927i)3-s + (−0.615 + 0.788i)4-s + (0.997 − 0.0697i)6-s + (−0.978 + 0.207i)7-s + (0.978 + 0.207i)8-s + (−0.719 − 0.694i)9-s + (−0.5 − 0.866i)12-s + (0.241 − 0.970i)13-s + (0.615 + 0.788i)14-s + (−0.241 − 0.970i)16-s + (−0.719 + 0.694i)17-s + (−0.309 + 0.951i)18-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (−0.559 + 0.829i)24-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (−0.374 + 0.927i)3-s + (−0.615 + 0.788i)4-s + (0.997 − 0.0697i)6-s + (−0.978 + 0.207i)7-s + (0.978 + 0.207i)8-s + (−0.719 − 0.694i)9-s + (−0.5 − 0.866i)12-s + (0.241 − 0.970i)13-s + (0.615 + 0.788i)14-s + (−0.241 − 0.970i)16-s + (−0.719 + 0.694i)17-s + (−0.309 + 0.951i)18-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (−0.559 + 0.829i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.626 + 0.779i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.626 + 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5456627645 + 0.2615683136i\)
\(L(\frac12)\) \(\approx\) \(0.5456627645 + 0.2615683136i\)
\(L(1)\) \(\approx\) \(0.6075554525 + 0.01075254804i\)
\(L(1)\) \(\approx\) \(0.6075554525 + 0.01075254804i\)

Euler product

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

−21.76361123852582040044632652057, −20.27359927936743918738678087463, −19.591419628292665886796160869172, −18.80900081718393419995547920325, −18.42007013092322276268033967098, −17.43302034125077069919352076114, −16.76103870345727923722264858356, −16.18440020542724563912606271069, −15.34671898272428558838203503773, −14.19860087693699147917443246555, −13.60084976916134100157065366346, −12.974423382965725972598375630717, −11.95676848304059764950778447388, −10.9925320257446044986200549558, −10.12121597791605811980026915616, −9.05622158126247920442614636119, −8.54838860099152343113764544435, −7.22440087782382940038361690126, −6.90040963293325097019013099410, −6.201779306746568313165636030653, −5.241228489518142012734414013587, −4.259688148330276077862863967310, −2.81914269272696523148609786234, −1.58473530056814708332121072929, −0.43745679289929881310193980057, 0.85320924310448303820848474044, 2.45002503793289663588579690390, 3.303472035112436069176686454421, 3.95413861919463017082728737180, 5.02726868860258921997343703241, 5.93283210172697725091396948300, 7.02263808179649737863079212518, 8.34089734671211766759212757004, 8.99825840614986658247882609616, 9.778747132620438727056323245626, 10.46449932486998600034174765713, 11.07367957200703847475583061213, 11.982762771761871192458515922677, 12.85117200500853889183458107569, 13.39280485098054173842079457395, 14.74188201954329696289541781055, 15.51535373190641920487965259088, 16.32502589723280841794300422860, 17.016495471137952967466534321699, 17.80061845357776533388875112515, 18.50999706813699215141986415625, 19.70824792300041785270410547528, 19.906444151035625279946648707765, 20.91763296769767817903502126357, 21.67913520425124477860324249113

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