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Properties

Label 1-1045-1045.459-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.898 - 0.438i$
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  + (−0.990 − 0.139i)2-s + (−0.241 + 0.970i)3-s + (0.961 + 0.275i)4-s + (0.374 − 0.927i)6-s + (0.913 − 0.406i)7-s + (−0.913 − 0.406i)8-s + (−0.882 − 0.469i)9-s + (−0.5 + 0.866i)12-s + (−0.848 + 0.529i)13-s + (−0.961 + 0.275i)14-s + (0.848 + 0.529i)16-s + (−0.882 + 0.469i)17-s + (0.809 + 0.587i)18-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (0.615 − 0.788i)24-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (−0.241 + 0.970i)3-s + (0.961 + 0.275i)4-s + (0.374 − 0.927i)6-s + (0.913 − 0.406i)7-s + (−0.913 − 0.406i)8-s + (−0.882 − 0.469i)9-s + (−0.5 + 0.866i)12-s + (−0.848 + 0.529i)13-s + (−0.961 + 0.275i)14-s + (0.848 + 0.529i)16-s + (−0.882 + 0.469i)17-s + (0.809 + 0.587i)18-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (0.615 − 0.788i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.898 - 0.438i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.898 - 0.438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02290744199 + 0.09913865672i\)
\(L(\frac12)\) \(\approx\) \(0.02290744199 + 0.09913865672i\)
\(L(1)\) \(\approx\) \(0.5120266722 + 0.1359075750i\)
\(L(1)\) \(\approx\) \(0.5120266722 + 0.1359075750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T \)
3 \( 1 + (-0.241 + 0.970i)T \)
7 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.848 + 0.529i)T \)
17 \( 1 + (-0.882 + 0.469i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.719 + 0.694i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.241 + 0.970i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.438 - 0.898i)T \)
53 \( 1 + (0.0348 + 0.999i)T \)
59 \( 1 + (-0.438 + 0.898i)T \)
61 \( 1 + (0.615 + 0.788i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.0348 + 0.999i)T \)
73 \( 1 + (-0.997 + 0.0697i)T \)
79 \( 1 + (-0.374 - 0.927i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.990 + 0.139i)T \)
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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

−20.77672987592070146387345146860, −20.150025927411815950767971451371, −19.28942062679491834291939683131, −18.716048720222550867287547858558, −17.82654457861278805149943756251, −17.50210175762162576216957284772, −16.77573120461876072186791310926, −15.644389224397630020824623923083, −14.91262455923563064467938544015, −14.13167708941924079288952971985, −13.00703856828487143695634538410, −12.17448371930222313474867634835, −11.38795595436392366051037397437, −10.91550557394698573981111722137, −9.73571014476509067240279951405, −8.77009723174347964849908196788, −8.14362110789739053259919512271, −7.30129496517062984944837582944, −6.744161465573202984975484973637, −5.56604693589425119092727861485, −4.97766987092974619392253051732, −3.048403931199027395831776365460, −2.15182701475601472913199046591, −1.418639144503078067089994111108, −0.05942279326938071903647576652, 1.51127278950647314969887585633, 2.52836128943765897400473473346, 3.70947066053099263093872266051, 4.614453766826549537854148064387, 5.53080004587645487276224507270, 6.7248494490254797105538837704, 7.47905705428272826663423426526, 8.60148630326277793885589974309, 9.0775736122961034035643230927, 10.05491656519901449278015299399, 10.74569596761034336149540040547, 11.34359637857605132047871627345, 12.03903725161063125962560902711, 13.23313058143176864688360735768, 14.68255068588831585946751615441, 14.87494608617606793137429961154, 15.90085081399575336282503884170, 16.91364604217027631800957152291, 17.03530996647815224210057833611, 17.97121879377820303781066829984, 18.77971444364568862104687039125, 19.98749894722338943669128587002, 20.17487841244686702660798160418, 21.277200617944155397069988485991, 21.59063553879193499733561206041

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