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Properties

Label 1-1053-1053.119-r0-0-0
Degree $1$
Conductor $1053$
Sign $0.923 - 0.384i$
Analytic cond. $4.89011$
Root an. cond. $4.89011$
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  + (0.998 − 0.0581i)2-s + (0.993 − 0.116i)4-s + (−0.727 − 0.686i)5-s + (−0.918 + 0.396i)7-s + (0.984 − 0.173i)8-s + (−0.766 − 0.642i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)14-s + (0.973 − 0.230i)16-s + (0.766 + 0.642i)17-s + (0.984 − 0.173i)19-s + (−0.802 − 0.597i)20-s + (−0.686 + 0.727i)22-s + (0.396 − 0.918i)23-s + (0.0581 + 0.998i)25-s + ⋯
L(s)  = 1  + (0.998 − 0.0581i)2-s + (0.993 − 0.116i)4-s + (−0.727 − 0.686i)5-s + (−0.918 + 0.396i)7-s + (0.984 − 0.173i)8-s + (−0.766 − 0.642i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)14-s + (0.973 − 0.230i)16-s + (0.766 + 0.642i)17-s + (0.984 − 0.173i)19-s + (−0.802 − 0.597i)20-s + (−0.686 + 0.727i)22-s + (0.396 − 0.918i)23-s + (0.0581 + 0.998i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.923 - 0.384i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ 0.923 - 0.384i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.348432163 - 0.4690237977i\)
\(L(\frac12)\) \(\approx\) \(2.348432163 - 0.4690237977i\)
\(L(1)\) \(\approx\) \(1.666328817 - 0.1804865285i\)
\(L(1)\) \(\approx\) \(1.666328817 - 0.1804865285i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.998 - 0.0581i)T \)
5 \( 1 + (-0.727 - 0.686i)T \)
7 \( 1 + (-0.918 + 0.396i)T \)
11 \( 1 + (-0.727 + 0.686i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (0.396 - 0.918i)T \)
29 \( 1 + (0.835 - 0.549i)T \)
31 \( 1 + (0.802 - 0.597i)T \)
37 \( 1 + (-0.342 - 0.939i)T \)
41 \( 1 + (0.448 + 0.893i)T \)
43 \( 1 + (0.686 + 0.727i)T \)
47 \( 1 + (-0.802 - 0.597i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.727 + 0.686i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (0.998 + 0.0581i)T \)
71 \( 1 + (-0.342 - 0.939i)T \)
73 \( 1 + (-0.984 + 0.173i)T \)
79 \( 1 + (-0.835 + 0.549i)T \)
83 \( 1 + (-0.549 - 0.835i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (0.957 + 0.286i)T \)
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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

−21.76032146847913151571046103503, −20.86041670700214421787206005133, −20.12189264084660357452677997595, −19.237203430101000289466047888, −18.8406305525783503783210634276, −17.61216767411804561709482511726, −16.44612096181463851347542663130, −15.87464486422171723446127755269, −15.49213524591283554077340273403, −14.20707829513695552378547091541, −13.87904728881306878300141856375, −12.93653983375978114495971126840, −12.06822438335890202943289131283, −11.42749168148847796597737209242, −10.51161964594367709772091560957, −9.9036266365745140352608695063, −8.409383142217811678762662251234, −7.39308679912977652030372952464, −7.00682903699451204092569052164, −5.937818665660874172217231384401, −5.15085676110901891032494597151, −4.00037363190758779722581275112, −3.07715595440915531389538382347, −2.89561999429618714264116007031, −1.020305898044660688520119461807, 0.91813665147221518203904033253, 2.359296874221942313165723531667, 3.18162786415190825120138132717, 4.1061297244070840922559564057, 4.94342616716978210728663724869, 5.71837844367373801230089316360, 6.68724950766377853592948411630, 7.59631755149672516418747806313, 8.355220752117726184798688529121, 9.63404727486169683277552233972, 10.31441326215159818947404829616, 11.47629213664592357313324483567, 12.16035157535824702713406822231, 12.80502203601513303238669202431, 13.27319888161507708661983357030, 14.48871378711015966765801981264, 15.2156032045952004483609910755, 16.000341704335105155669582020654, 16.30813321488212743751232352727, 17.38813524399753669231959842748, 18.631208814341101051128844114175, 19.43369159301398162963960622679, 19.95982096086931837843340095181, 20.88387622023031863679644351742, 21.32581421357585058275367812630

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