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Properties

Label 1-1045-1045.234-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.533 + 0.845i$
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  + (0.882 + 0.469i)2-s + (0.997 + 0.0697i)3-s + (0.559 + 0.829i)4-s + (0.848 + 0.529i)6-s + (0.104 − 0.994i)7-s + (0.104 + 0.994i)8-s + (0.990 + 0.139i)9-s + (0.5 + 0.866i)12-s + (0.374 + 0.927i)13-s + (0.559 − 0.829i)14-s + (−0.374 + 0.927i)16-s + (−0.990 + 0.139i)17-s + (0.809 + 0.587i)18-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (0.0348 + 0.999i)24-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.997 + 0.0697i)3-s + (0.559 + 0.829i)4-s + (0.848 + 0.529i)6-s + (0.104 − 0.994i)7-s + (0.104 + 0.994i)8-s + (0.990 + 0.139i)9-s + (0.5 + 0.866i)12-s + (0.374 + 0.927i)13-s + (0.559 − 0.829i)14-s + (−0.374 + 0.927i)16-s + (−0.990 + 0.139i)17-s + (0.809 + 0.587i)18-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (0.0348 + 0.999i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.533 + 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.406008177 + 1.879040947i\)
\(L(\frac12)\) \(\approx\) \(3.406008177 + 1.879040947i\)
\(L(1)\) \(\approx\) \(2.328876086 + 0.8141207157i\)
\(L(1)\) \(\approx\) \(2.328876086 + 0.8141207157i\)

Euler product

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

−21.52394324855077659188651078555, −20.508956972213431675868513334902, −20.14703567738062261518081152507, −19.27049439912870428698846887656, −18.51825820069756809047175609837, −17.90885446508378722292384958694, −16.31682264953552390856857528364, −15.51710540874341521578803117182, −15.029384009525273460295521714502, −14.37980328677418584566055332750, −13.358258692207499394868637258352, −12.882019743612956259100752663860, −12.145064404049566008943807648224, −11.072935858116084780307917619467, −10.37554087252309393410720983877, −9.21945536339121565730986247977, −8.73635484327639540301383593697, −7.56126690720272356549041597816, −6.6434292314418522837827641389, −5.64924206477942174061234222417, −4.79203429366663970486567350580, −3.79795979418971809463371709875, −2.86152746404739331804354125848, −2.327452799065323321332926068476, −1.20608916275585568022118173831, 1.512825244617730655871445037923, 2.49966908071201223691662744647, 3.557625374680720086225382280260, 4.20151885091564958604560137725, 4.88687741905709598820882328458, 6.331458155088603894711346811709, 7.022003782543820823953329306218, 7.71502391008098197381963819768, 8.62388631570158987645311609341, 9.413994620183239579176411237739, 10.6493597065079936223307081164, 11.33002118438937625221395201567, 12.468264687707628924720222989870, 13.49305534535153946734475572, 13.60126095156372031526566291597, 14.53152884127197475367266949292, 15.22894718859700030417545894505, 15.99602528188541197783845863303, 16.77558211540829883358282074891, 17.55184975177345176056893011051, 18.67967510116179364914903266382, 19.64810096538218053280962712499, 20.24307800044158476772550036883, 20.975019873456852852336837682860, 21.55252635572267471382969793507

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