Properties

Label 1-1045-1045.593-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.558 - 0.829i$
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  + (−0.642 + 0.766i)2-s + (−0.342 − 0.939i)3-s + (−0.173 − 0.984i)4-s + (0.939 + 0.342i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.766 + 0.642i)9-s + (−0.866 + 0.5i)12-s + (−0.342 + 0.939i)13-s + (−0.173 + 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.642 + 0.766i)17-s i·18-s + (−0.766 − 0.642i)21-s + (−0.984 + 0.173i)23-s + (0.173 − 0.984i)24-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.342 − 0.939i)3-s + (−0.173 − 0.984i)4-s + (0.939 + 0.342i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.766 + 0.642i)9-s + (−0.866 + 0.5i)12-s + (−0.342 + 0.939i)13-s + (−0.173 + 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.642 + 0.766i)17-s i·18-s + (−0.766 − 0.642i)21-s + (−0.984 + 0.173i)23-s + (0.173 − 0.984i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.558 - 0.829i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.558 - 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2153815286 - 0.4046516458i\)
\(L(\frac12)\) \(\approx\) \(0.2153815286 - 0.4046516458i\)
\(L(1)\) \(\approx\) \(0.6071070288 - 0.06862997235i\)
\(L(1)\) \(\approx\) \(0.6071070288 - 0.06862997235i\)

Euler product

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

−21.70984983439237720280062839301, −21.0189552283046803205033720482, −20.20774215468302300233475955992, −19.833217452020574194009296076961, −18.49163653283864948975178446280, −17.82036333587138135268150685158, −17.46545919278395703105375058836, −16.3261817533127348375544490755, −15.79125492665895635899174472755, −14.82062398964036350400590217740, −13.9978001690594129388827716354, −12.79622546853459360271306274682, −11.9459828348483004177857196165, −11.390937015857880962289994797149, −10.56574571090023759568397689134, −9.96341887709906440463161732814, −9.012279923107078241425316661660, −8.4044475619085083064413094757, −7.50268960760173217186692048475, −6.19581145484631885905558056667, −4.98563890570427677530298728849, −4.53072083962024095873507530588, −3.26494162249182913045194650672, −2.55816684431242367591200858430, −1.26088688327159970806662402862, 0.2615217631426757954886882467, 1.6371436961273354351826481310, 2.10957638408184681050201501162, 4.12502269238787129989649588614, 4.9699052196451038589316161862, 6.00903995754423411999577034210, 6.65094208331178643990872291935, 7.55068420000880813370014674976, 8.072438789663914810146263806429, 8.926691541940399445211543555004, 9.99520725996817780250258645821, 10.98837676600500585267524197839, 11.49948276681966606345056038247, 12.58927438752675642975481773799, 13.696482532928823897771728969161, 14.12692995881353231238410134529, 14.944099383873338900316029269958, 15.98999247546160001337376706250, 16.90666112746661727543680899400, 17.33518438829474529592998467733, 18.0629903109137309751529467314, 18.70063739972309335711740591282, 19.628630102207555853260990404799, 20.0308319851262583104497444169, 21.30330793726618783296603831862

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