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Properties

Label 1-1045-1045.282-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.972 - 0.231i$
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  + (0.469 + 0.882i)2-s + (0.0697 + 0.997i)3-s + (−0.559 + 0.829i)4-s + (−0.848 + 0.529i)6-s + (−0.994 + 0.104i)7-s + (−0.994 − 0.104i)8-s + (−0.990 + 0.139i)9-s + (−0.866 − 0.5i)12-s + (−0.927 − 0.374i)13-s + (−0.559 − 0.829i)14-s + (−0.374 − 0.927i)16-s + (0.139 − 0.990i)17-s + (−0.587 − 0.809i)18-s + (−0.173 − 0.984i)21-s + (0.342 + 0.939i)23-s + (0.0348 − 0.999i)24-s + ⋯
L(s)  = 1  + (0.469 + 0.882i)2-s + (0.0697 + 0.997i)3-s + (−0.559 + 0.829i)4-s + (−0.848 + 0.529i)6-s + (−0.994 + 0.104i)7-s + (−0.994 − 0.104i)8-s + (−0.990 + 0.139i)9-s + (−0.866 − 0.5i)12-s + (−0.927 − 0.374i)13-s + (−0.559 − 0.829i)14-s + (−0.374 − 0.927i)16-s + (0.139 − 0.990i)17-s + (−0.587 − 0.809i)18-s + (−0.173 − 0.984i)21-s + (0.342 + 0.939i)23-s + (0.0348 − 0.999i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.972 - 0.231i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (282, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.972 - 0.231i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3385341271 - 0.03976354554i\)
\(L(\frac12)\) \(\approx\) \(0.3385341271 - 0.03976354554i\)
\(L(1)\) \(\approx\) \(0.6129051383 + 0.5761310385i\)
\(L(1)\) \(\approx\) \(0.6129051383 + 0.5761310385i\)

Euler product

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

−21.619750792811792331497147944372, −20.67893499195435339222606478620, −19.83665138175659142281105889478, −19.215013722673636910197646807248, −18.93601149616032570683993416304, −17.83519468412690489589555736024, −17.12893432063497685046815133358, −16.1179954494137541278639468805, −14.83131341653727279123802259972, −14.37674163924676993833014693516, −13.374237822264393340105499113079, −12.7664447453100705885319658388, −12.304307322313184719488278314328, −11.41941761000220605780207612033, −10.47843285065316336358691007999, −9.63620183014450245831087435150, −8.82875142782517977163277672130, −7.78542854398882830465414106007, −6.62034225563408789316666837678, −6.14975916613759329131465291535, −5.03594058517870476353701607089, −3.93514278586490124912199190031, −2.946721289947431586773516132454, −2.255910461656120112199683908333, −1.16399684360673662937034814068, 0.12636341664764233659162396304, 2.63651237381814314166317438972, 3.31323590025297523133147111914, 4.15573232877316242254412765162, 5.255329917410672297978403446559, 5.6105427836505955524277144394, 6.87470102368105549925492703793, 7.52082058784886523020608102375, 8.73639500193339510711060605519, 9.386795234752968748811108477665, 10.00150864703259961481830565496, 11.17597268036149648750443115790, 12.19402162960555266339623284772, 12.8905495745140260077621800376, 13.91211272432566196906848032304, 14.51472241691006584258009279402, 15.41577478833610163931442586165, 15.9212798940797807362904440628, 16.582988114134845536613799512766, 17.294951923309387990612500721187, 18.14815809775693817099118333542, 19.29234790159939905607961938046, 20.072535962915070118368337227117, 20.95369452995652411483043544143, 21.83668967677102372323263093229

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