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Properties

Label 1-1045-1045.558-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.999 + 0.0361i$
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  + (−0.743 + 0.669i)2-s + (0.994 − 0.104i)3-s + (0.104 − 0.994i)4-s + (−0.669 + 0.743i)6-s + (0.587 − 0.809i)7-s + (0.587 + 0.809i)8-s + (0.978 − 0.207i)9-s − i·12-s + (0.207 + 0.978i)13-s + (0.104 + 0.994i)14-s + (−0.978 − 0.207i)16-s + (0.207 − 0.978i)17-s + (−0.587 + 0.809i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (0.669 + 0.743i)24-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (0.994 − 0.104i)3-s + (0.104 − 0.994i)4-s + (−0.669 + 0.743i)6-s + (0.587 − 0.809i)7-s + (0.587 + 0.809i)8-s + (0.978 − 0.207i)9-s − i·12-s + (0.207 + 0.978i)13-s + (0.104 + 0.994i)14-s + (−0.978 − 0.207i)16-s + (0.207 − 0.978i)17-s + (−0.587 + 0.809i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (0.669 + 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.999 + 0.0361i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (558, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.999 + 0.0361i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.732882021 + 0.03132025960i\)
\(L(\frac12)\) \(\approx\) \(1.732882021 + 0.03132025960i\)
\(L(1)\) \(\approx\) \(1.190138988 + 0.1192500411i\)
\(L(1)\) \(\approx\) \(1.190138988 + 0.1192500411i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.743 + 0.669i)T \)
3 \( 1 + (0.994 - 0.104i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (0.207 + 0.978i)T \)
17 \( 1 + (0.207 - 0.978i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.406 - 0.913i)T \)
53 \( 1 + (-0.207 - 0.978i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.406 - 0.913i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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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

−21.2338913858222976983561129831, −20.77812902353669952234755497037, −19.94254352468977595173265407372, −19.13068818011384097861363361092, −18.74713848769737796538981065115, −17.68846318121989667418857458523, −17.23922848768731032182202632131, −15.79583304078857465068530057866, −15.46221429104415137120628307836, −14.48010715851053443111017654537, −13.52796721877518063786125031274, −12.655502200749137135155638846628, −12.11865970971211793389787493944, −10.84979254169807517124287356535, −10.416592247208358757264594606936, −9.23100119468350066095277974690, −8.82701522721529502467494217270, −7.936884248175674543923699578103, −7.45358147395102423811225358168, −6.02993471529950106469373326289, −4.797182298981078416006778986306, −3.72183626677215487459279312821, −2.94125747273420891875942814503, −2.090391437627111697686899028738, −1.20608012924364506568158332473, 0.99559845898740640492944528172, 1.82293952771415366164581917061, 2.9851109907170670340017043532, 4.304945557662079632198833892164, 4.95289385250516246085767353762, 6.410796284683736763363921850540, 7.17003661905625701955332993940, 7.71359743675064123382453444907, 8.66240180404231065882320078885, 9.24933196951624795468952013206, 10.10264481844755051456877748966, 10.93523610763515007370145678998, 11.85554462832449207504094276201, 13.308978796793417603740765987835, 13.84644404502243086051796869495, 14.584351083913181339443220160828, 15.13227525888665816327881488113, 16.269962551226112063082283103013, 16.67719277329707206731259757601, 17.76436159521132077593709341021, 18.46870311829060417144708485236, 19.104817282054441676393432463581, 19.89776991217723350292122692788, 20.64964346198194045856270309104, 21.10897745304344624830769398277

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