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Properties

Label 1-1045-1045.658-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: $\chi_{1045} (658, \cdot )$
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

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

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