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Properties

Label 1-1009-1009.147-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.938 + 0.345i$
Analytic cond. $4.68577$
Root an. cond. $4.68577$
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  + (0.840 − 0.542i)2-s + (0.294 − 0.955i)3-s + (0.411 − 0.911i)4-s + (−0.246 + 0.969i)5-s + (−0.270 − 0.962i)6-s + (0.124 − 0.992i)7-s + (−0.149 − 0.988i)8-s + (−0.826 − 0.563i)9-s + (0.318 + 0.947i)10-s + (−0.510 − 0.859i)11-s + (−0.749 − 0.661i)12-s + (−0.258 + 0.965i)13-s + (−0.433 − 0.900i)14-s + (0.853 + 0.521i)15-s + (−0.661 − 0.749i)16-s + (0.0622 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.840 − 0.542i)2-s + (0.294 − 0.955i)3-s + (0.411 − 0.911i)4-s + (−0.246 + 0.969i)5-s + (−0.270 − 0.962i)6-s + (0.124 − 0.992i)7-s + (−0.149 − 0.988i)8-s + (−0.826 − 0.563i)9-s + (0.318 + 0.947i)10-s + (−0.510 − 0.859i)11-s + (−0.749 − 0.661i)12-s + (−0.258 + 0.965i)13-s + (−0.433 − 0.900i)14-s + (0.853 + 0.521i)15-s + (−0.661 − 0.749i)16-s + (0.0622 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.938 + 0.345i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.938 + 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2893271372 - 1.625426019i\)
\(L(\frac12)\) \(\approx\) \(-0.2893271372 - 1.625426019i\)
\(L(1)\) \(\approx\) \(0.9739354042 - 1.054222481i\)
\(L(1)\) \(\approx\) \(0.9739354042 - 1.054222481i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (0.840 - 0.542i)T \)
3 \( 1 + (0.294 - 0.955i)T \)
5 \( 1 + (-0.246 + 0.969i)T \)
7 \( 1 + (0.124 - 0.992i)T \)
11 \( 1 + (-0.510 - 0.859i)T \)
13 \( 1 + (-0.258 + 0.965i)T \)
17 \( 1 + (0.0622 - 0.998i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (-0.399 + 0.916i)T \)
29 \( 1 + (-0.603 - 0.797i)T \)
31 \( 1 + (0.210 + 0.977i)T \)
37 \( 1 + (0.342 - 0.939i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + (0.999 + 0.0373i)T \)
53 \( 1 + (-0.951 + 0.306i)T \)
59 \( 1 + (0.943 - 0.330i)T \)
61 \( 1 + (0.467 - 0.884i)T \)
67 \( 1 + (-0.124 - 0.992i)T \)
71 \( 1 + (-0.840 + 0.542i)T \)
73 \( 1 + (-0.757 - 0.652i)T \)
79 \( 1 + (0.234 + 0.972i)T \)
83 \( 1 + (-0.445 - 0.895i)T \)
89 \( 1 + (-0.489 + 0.872i)T \)
97 \( 1 + (0.773 - 0.633i)T \)
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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

−22.12916345765191786703108014093, −21.29788901302727496420596203687, −20.666565210487597160242488421483, −20.18503684724285656163825634208, −19.17892817416831656404243935623, −17.75986351423098640105778327950, −17.150774176226988805812362256, −16.28167193450705724720137509292, −15.64019330263864541834213710009, −14.9324513521596737569932636658, −14.67387342701465272114029261337, −13.108507895499014523078318405809, −12.77313185677867122900787961526, −11.95683602718599453505120640936, −10.94726818160092864867342205538, −9.96696763578766388913851870256, −8.862481496180978711300841444, −8.31973715185550949160546173991, −7.626195531099677394571071337484, −6.05303934722206330914874245313, −5.46395358963573349436342722073, −4.61479712908838920988667870667, −4.11346095203911169381604382050, −2.850472638340828228375312334218, −2.09953812909720945609467464952, 0.45294936618059634879015580435, 1.792861006820385381846811482367, 2.611595273741325440019068691003, 3.494323453589588440412208217841, 4.251494970743213106293915386382, 5.625028059896561620369657032511, 6.45325078893847584685459868734, 7.19195420875961729809353660333, 7.79950129117094557868812715476, 9.17967565346476362335084236377, 10.216293342662692538056472149752, 11.14501615573587704258572863429, 11.497860834042683238293940599461, 12.51466778494858995436854490972, 13.43695004404091012328078017144, 14.11002647506394151352458360591, 14.288327902857886726208165614890, 15.44841517073902418602465850291, 16.358183624000646376606427048822, 17.492372226159648980800983691500, 18.40366706456820333472372286259, 19.18173849191833279311408652743, 19.40967268004737435567940981585, 20.44858718966235489504799235822, 21.17508869003377432126687926046

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