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Properties

Label 1-1045-1045.139-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.289 + 0.957i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.289 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.732767132 + 3.511520647i\)
\(L(\frac12)\) \(\approx\) \(4.732767132 + 3.511520647i\)
\(L(1)\) \(\approx\) \(2.389454049 + 0.8050482743i\)
\(L(1)\) \(\approx\) \(2.389454049 + 0.8050482743i\)

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

−21.03879157633417437928791169316, −20.557613040898719923458373155541, −19.79899995096862652675275382527, −19.013882060209940782180809078120, −17.86886576957500473606726685297, −17.43589980200045888049610529562, −16.46752558064044393630435404289, −15.33564842490499150902165360538, −14.81294400603239044073465432880, −13.8775998107427299068847940517, −13.387379381511434780413709486409, −12.709075127516838930744505649, −11.71625189564507848469578246403, −11.1690046357559453187233223487, −10.30246736864828594832345972516, −8.492292891217635909631618990271, −8.25878516618078613450537363524, −7.10623439051975761128911110243, −6.55164171704481902278182836862, −5.59322711517513095144047065906, −4.672463822070904393436313738686, −3.67917971109490606959525223460, −2.68629523377868005361270042460, −1.7555200414651756432371490395, −0.84296850224340458282463700495, 1.2578025499726734644613465366, 2.40261502197290490511486042951, 3.17681199081771596156534298255, 4.34407736427675587293649542941, 4.71239390924020507724275004151, 5.67064353482131587301354250857, 6.538811112902967338865224661, 7.74474645217486494132459155410, 8.671757772097767511177666442126, 9.49223267517735225281109779736, 10.61879079504292082699068151963, 11.30561550867018885194251644985, 11.67113132087461242312247518646, 12.99177661329901018959814429211, 13.7576757286369984499754505838, 14.481266727414475953108732008620, 15.12636516763430375681433498904, 15.80650592712514117144891594428, 16.47257540045992268115681183941, 17.45391694453813513720772685443, 18.48218034291720324446928010868, 19.51398062402532229628903373798, 20.28636460534963505563007202516, 20.940835346831008496234062175756, 21.49082774316693059171367844626

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