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Properties

Label 1-1045-1045.284-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.935 + 0.352i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 12-s + (−0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)18-s − 21-s − 23-s + (−0.809 − 0.587i)24-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 12-s + (−0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)18-s − 21-s − 23-s + (−0.809 − 0.587i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.935 + 0.352i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1417064531 + 0.02578031914i\)
\(L(\frac12)\) \(\approx\) \(0.1417064531 + 0.02578031914i\)
\(L(1)\) \(\approx\) \(0.4763859789 - 0.4132579743i\)
\(L(1)\) \(\approx\) \(0.4763859789 - 0.4132579743i\)

Euler product

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

−21.39643517391259980888526335938, −20.349656435860380635466051007731, −19.638285324721060980031057402066, −18.97660832769859977408607879910, −18.22751652756116697139289763499, −17.14995013562587410586571229178, −16.589555835213780460746268169060, −15.861065668637064361364624672889, −15.15199499182399999553364341094, −14.56265655755856425689033240157, −13.83084677883912563054008645762, −12.37189329754071880209401261799, −11.5912530683036102128618687901, −10.57763070457280405729007590718, −9.818549925816405091470073699887, −9.29846785122359092848286695243, −8.47283662009218466003461711739, −7.77291206242650505324779272265, −6.596872259359686313011116083958, −5.67121048480624688511491869058, −5.05183008117473423154509071593, −3.86994172855223415000997933816, −2.65253645865161263769928848927, −1.81570863191000468656002426185, −0.04767411089830656889704890225, 0.77774320850131672694085710013, 1.70327380493413085765297020166, 2.83202189772496299989471326642, 3.42980656982809694556653710444, 4.70873997573851718489891357162, 6.22614516715619693256647273552, 6.97415923910291484254602069474, 7.91917342315282589139030297707, 8.1009689870121502115405812051, 9.63762575946671074666737488739, 9.89290211463490528691415381754, 11.04907630393544418802223650639, 11.9269247748173505443685911519, 12.51295261480240816691427519265, 13.3956439391485631566706283350, 13.99286276019977933255137348002, 15.09259958733076254778993881714, 16.2483463764343660835271750937, 17.003702973676694705735266919560, 17.606859598684029846802177804287, 18.36160376816659582779064928714, 19.14827157449927216513988462243, 19.77065631177693823109971486558, 20.32609469261234012167159268506, 21.00879398601125031825772799468

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