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Properties

Label 1-1045-1045.27-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.973 - 0.229i$
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  + (0.406 − 0.913i)2-s + (0.743 + 0.669i)3-s + (−0.669 − 0.743i)4-s + (0.913 − 0.406i)6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (0.104 + 0.994i)9-s − i·12-s + (0.994 − 0.104i)13-s + (0.669 − 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.994 − 0.104i)17-s + (0.951 + 0.309i)18-s + (0.5 + 0.866i)21-s + (0.866 + 0.5i)23-s + (−0.913 − 0.406i)24-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (0.743 + 0.669i)3-s + (−0.669 − 0.743i)4-s + (0.913 − 0.406i)6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (0.104 + 0.994i)9-s − i·12-s + (0.994 − 0.104i)13-s + (0.669 − 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.994 − 0.104i)17-s + (0.951 + 0.309i)18-s + (0.5 + 0.866i)21-s + (0.866 + 0.5i)23-s + (−0.913 − 0.406i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.515426314 - 0.2931703125i\)
\(L(\frac12)\) \(\approx\) \(2.515426314 - 0.2931703125i\)
\(L(1)\) \(\approx\) \(1.674546388 - 0.3203450257i\)
\(L(1)\) \(\approx\) \(1.674546388 - 0.3203450257i\)

Euler product

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

−21.41588849620561591080128824297, −20.90259889955774915164445071796, −20.165736354252524194574621248628, −19.022270428300675042262022922057, −18.35145590592779272740596030959, −17.578546896144727950429730362683, −17.0729133975285291726859656193, −15.752580556701494625299744502617, −15.284089209872855292710480955704, −14.38713419539045594598788414878, −13.72801343512256702201173875050, −13.28649563951866991448535156876, −12.268367035140925901277859873750, −11.46811567398040927430182465570, −10.31072476868774799967045361057, −8.863183820164200072109277989726, −8.60206107643270486591720464514, −7.77702447133436229488674238807, −6.86475436688065199519393246955, −6.32564732556827216861167775180, −5.075631931857332509582205869777, −4.226129984048365062693567852671, −3.34203851634386019103684919616, −2.20520340541806467541631177449, −0.97575035273148252421906514732, 1.3067214362399911850348365048, 2.19825279701607810656798844054, 3.111885738566956976733524459148, 3.98192755373308564086776836763, 4.81655603162671623409521220513, 5.44983421466574290174627649052, 6.77537367876686591182484876447, 8.28487887228285721053801298357, 8.646508232304734860454936126914, 9.522580078187147582733914801619, 10.48651851810963364899290844458, 11.11030456410368801009222294946, 11.76054787716934751252413388207, 12.99019426451905332391033651369, 13.64701436187740637836447178611, 14.268058941470746757207492602380, 15.2741688433639318492064482779, 15.49872045548193187697386987961, 16.862464697749757235781690871717, 17.93246062668617139246246787028, 18.51724195735331489908277194379, 19.46476880828309641245018026610, 20.13310855037519620803433100705, 20.83190662636519971800965180004, 21.35470760002942951641798145256

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