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Properties

Label 1-1045-1045.1018-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.979 - 0.203i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.979 - 0.203i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1018, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.979 - 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1094450846 + 1.064729658i\)
\(L(\frac12)\) \(\approx\) \(-0.1094450846 + 1.064729658i\)
\(L(1)\) \(\approx\) \(0.6727981281 + 0.6884413080i\)
\(L(1)\) \(\approx\) \(0.6727981281 + 0.6884413080i\)

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

−20.93497682256577347972940032389, −20.07186689100620756288765153493, −19.81743192186963500561535209926, −18.878295805110527311731958340803, −18.0952714508425853973558094975, −17.54449335868163418904786935031, −16.86753432303463692422456313281, −15.42806406741065347677036334099, −14.64287977387370521006763834234, −13.6997855254427785536653706275, −13.360473734237449887736781397434, −12.10213461097950719790548398132, −11.84102312833367423958211043466, −10.69396253002175450064725338092, −9.89879791749110952345157148622, −8.98274021808538362685801605126, −8.252357286101511165132337931431, −7.59737804992356092979059563146, −6.79406251202274211870656228815, −5.23611732590970592732618601519, −4.25056947633606619896892564151, −3.42093595232118253899006418498, −2.142774453051277235727388448053, −1.882478579842194603742907213367, −0.44122387357034243973010759328, 1.611839110932302914199516171963, 2.52734431710335087926095695981, 3.98165856902893132373366278912, 4.81790104207911244784924312651, 5.31628224864308726523178573640, 6.676513264119060553710076380122, 7.50579765556961016882296867922, 8.399897358120186270970121518064, 8.81422348443092505435997049339, 9.7833355836915014755028480334, 10.47634111828716640454141267513, 11.38282956489362818454880853507, 12.63691398100886032549837693310, 13.7226813969388080011450308305, 14.42190711165703603085658829290, 14.8391210933463814367796044769, 15.68721856197241690577801372172, 16.32137439029353211737916113822, 17.238348882342415278147479694653, 17.97268597056164275879281328615, 18.69964858602958752217098459310, 19.815459994393269784380673126493, 20.06561442510295816238754684125, 21.33160654590343133162414452015, 21.92445515743653466796670158932

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