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Properties

Label 1-1045-1045.142-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.991 - 0.128i$
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  + (0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.939 + 0.342i)9-s + (0.866 − 0.5i)12-s + (−0.984 + 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s + i·18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (0.766 + 0.642i)24-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.939 + 0.342i)9-s + (0.866 − 0.5i)12-s + (−0.984 + 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s + i·18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (0.766 + 0.642i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.991 - 0.128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4245938352 + 0.02737141461i\)
\(L(\frac12)\) \(\approx\) \(0.4245938352 + 0.02737141461i\)
\(L(1)\) \(\approx\) \(0.5541360317 + 0.2958053446i\)
\(L(1)\) \(\approx\) \(0.5541360317 + 0.2958053446i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 + (-0.984 - 0.173i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.342 - 0.939i)T \)
53 \( 1 + (-0.642 - 0.766i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.342 - 0.939i)T \)
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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

−21.80028186787236756628038784406, −20.80453211330926884670751662738, −20.06354445550664571742956096393, −19.30959975916698872049507256183, −18.5283546630820820339150158846, −17.730404980914091638311849489819, −16.988041743158221543690098925329, −16.133996292621316157237164245774, −15.32977752422505488679543197854, −14.236707859613517502508164352111, −13.491568695411156549636745245919, −12.458429411108007403151718728215, −12.236188159531214448003852542650, −11.14873971055363531851239578020, −10.51874984729461697040943713188, −9.68175356022955235516729918053, −9.26652913359664669724587480515, −7.56316306561789272137639825580, −6.77226182458267756390149283640, −5.63394682256129594967172911244, −5.11330994660406268735538021419, −4.027245410134489192000912832443, −3.324528808075951556368609741348, −2.09537410689107396337642369702, −0.82423331965590701475206464249, 0.24810914599732788293584023525, 2.064242866946971443228883950309, 3.41254697827587930729680126154, 4.3972279551967785027958510113, 5.264986307130348146518979499091, 6.08389747661380257584234357032, 6.58315647122193924157687534150, 7.51811080630088473083128364028, 8.36773827377559635671675650002, 9.61109046164953153907683901232, 10.05400380432086693453202385959, 11.44028299450200333227062616356, 12.250572503049380384142637563789, 12.80388217773984527763580510811, 13.47532536408564213586775113671, 14.751689727875627096681213842907, 15.23139141862768238022684514054, 16.267361098626749748454147698539, 16.73014595299838329630985047403, 17.324489131597706164923327828900, 18.39770150407224314935772397127, 18.78597379163569386522355518620, 19.8210877271305074944623404376, 21.18081126995413949567873012286, 21.97928049180247558886075911563

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