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Properties

Label 1-1045-1045.1019-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.783 + 0.621i$
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  + (0.978 + 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (0.978 − 0.207i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + 12-s + (−0.669 + 0.743i)13-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (0.809 − 0.587i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.978 + 0.207i)24-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (0.978 − 0.207i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + 12-s + (−0.669 + 0.743i)13-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (0.809 − 0.587i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.978 + 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.783 + 0.621i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.546676356 + 1.235333434i\)
\(L(\frac12)\) \(\approx\) \(3.546676356 + 1.235333434i\)
\(L(1)\) \(\approx\) \(2.383259808 + 0.4012193438i\)
\(L(1)\) \(\approx\) \(2.383259808 + 0.4012193438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.104 - 0.994i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−21.43107912423466356294384552762, −20.62195180205938571660096542448, −20.12912541242504259295748619128, −19.39363200011595755116274547878, −18.820689501605304855899004932652, −17.39089277547348727675696767006, −16.28116056082619804471234206127, −15.938205656491993632973679753184, −14.98027993335210744217499506241, −14.26552659299514862536468231298, −13.717007305242530332976845902452, −12.78147474126096415743403102573, −12.32515013837940565920123939250, −10.98250120291490960400188663283, −10.19783842694503118958609310187, −9.73997993039679983414638970917, −8.53293380590285799048529691575, −7.38989223565822565003606800273, −6.92729472243813045175315741456, −5.637200892310960023607980853643, −4.756482516343312417858307283229, −3.92512273907973174595895574616, −3.01007220767159515452429603163, −2.56434807772268325698472889310, −1.06113479299507605218202883259, 1.564275435126215287206376026116, 2.46994172934029866581089639971, 3.26666548023163059451939637982, 4.00821774319322662210183972867, 5.16393012389077101850717752641, 6.16596674248768359178354725874, 6.89254342265620914671949095808, 7.65476638297425700200286604898, 8.61048173120913246674952767473, 9.49567224605937857862271837003, 10.381972289379856807002895932617, 11.79843749397828974917766027511, 12.23521486930201700081321184372, 13.1158429012356074766071533160, 13.623759706506791596971041994257, 14.68052402713281737929955138406, 14.99987365610635543181055600030, 15.94527765015631902590901415881, 16.66889624848034687642321065368, 17.68801500176715780146142809083, 18.907116236461932910935428353608, 19.38745421585505966575678367240, 20.01718618104272831782884724950, 21.14291809176717393464600959875, 21.48438924873460745985059939114

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