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Properties

Label 1-1045-1045.863-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.229 - 0.973i$
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

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

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3165132815 - 0.2505528879i\)
\(L(\frac12)\) \(\approx\) \(0.3165132815 - 0.2505528879i\)
\(L(1)\) \(\approx\) \(0.5225312103 + 0.04231531413i\)
\(L(1)\) \(\approx\) \(0.5225312103 + 0.04231531413i\)

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

−21.69892014083144960734718550888, −21.066087938699398425364098970574, −20.122412550370128154309599593972, −19.40425283820668645294512091771, −18.69925370061254309744334807362, −17.75229916829315823848178923312, −17.160164437997304208286256776803, −16.3515646521251219143380651834, −15.736157338852351798197446761191, −14.63452331958875009581489385840, −13.64381907888421306847632879894, −12.52027609266561755405960411602, −12.14786509593148510712936577202, −11.41320468916277602869937784524, −10.24967078807588186219893534679, −9.92168084991641963648252272611, −9.27341515237949984845102565799, −8.162983904106190885551491259164, −7.108793037695325549140172553128, −5.99883976700624247928789697994, −5.11617838402706615083820603636, −4.15993505066504766980227574613, −3.298949112768584868654809757576, −2.44666314018277605552704811851, −0.93188267467405698987547105391, 0.291906787316538205898690480266, 1.41104096623990704447939807510, 2.797679446391355227995076318662, 4.29432962221109510587199057739, 5.13097280319389876077214684527, 6.161914540591510475812398750140, 6.54991151159712286661522178653, 7.56555258882522437494578633587, 8.03672497533540091807940610823, 9.39808258951776481495652930093, 10.04725284825909728346735578128, 10.76811665080669608182106891333, 12.08269251209594978514444285717, 12.63718649061917358129228052046, 13.60051871788678767725875670286, 14.24406965364938942413438370188, 15.24468677104830204031715739457, 16.3322629325343448874581498678, 16.56782272505571330207912099909, 17.40186413968998672581760193688, 18.11256074155679138194760011249, 19.02648170868739013554216997039, 19.35820221946213425809176261785, 20.30845349165930717526875954524, 21.86277733416019330174963194735

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