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Properties

Label 1-1045-1045.843-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.865 + 0.501i$
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  + (0.207 − 0.978i)2-s + (0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.669 + 0.743i)9-s − i·12-s + (−0.743 − 0.669i)13-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (−0.743 + 0.669i)17-s + (0.587 + 0.809i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.978 − 0.207i)24-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.669 + 0.743i)9-s − i·12-s + (−0.743 − 0.669i)13-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (−0.743 + 0.669i)17-s + (0.587 + 0.809i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.978 − 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.865 + 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.085949120 + 0.2920251673i\)
\(L(\frac12)\) \(\approx\) \(1.085949120 + 0.2920251673i\)
\(L(1)\) \(\approx\) \(0.9873712022 - 0.1563071030i\)
\(L(1)\) \(\approx\) \(0.9873712022 - 0.1563071030i\)

Euler product

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

−21.7055892891165019996620682019, −20.77330922228898625641905808630, −19.57147977885284903653625085280, −19.026890586387812497588794324143, −18.35065903189425457576994140873, −17.542584017563708545084732644356, −16.86334791639935049725472027814, −15.814347683992875667347880618991, −15.20805539147311414344091948396, −14.38964359356139648728319083927, −13.62338395049232981197743149840, −12.98014478600960181787570519415, −12.188128653178765314834314539890, −11.526632275968534499482813951398, −9.79545204717727296004013933876, −9.10736852285923754375286908726, −8.53460883137164971031542219674, −7.42698122534751223528634716013, −6.89709575359428652770851437516, −6.098927189686014855780108281568, −5.25740797010315187853385088969, −4.14138212768540018797543212301, −2.965658377047491137820491140825, −2.19698271150190215987097252191, −0.477670086293394723174996050816, 1.07742566889520614132021260537, 2.61133206342658218641500464388, 3.0947260135360591223545422110, 4.174817131485471230856018606201, 4.68692405962036231612246060193, 5.72024030276200719584071750639, 6.951110531508541963373722732131, 8.227876005584392996294319014426, 8.92776176809940930153190441600, 9.87068314395343459964320078001, 10.41622303633700338479047373651, 10.934755211696114702150930201617, 12.07368919433450894379266475359, 12.98381578662706812953712072567, 13.59525074598723556216810283012, 14.49664609811812651042798174579, 15.1161453745477418047142408941, 16.06371868670057777486051260068, 17.07518609076760437460572698243, 17.59538663529233208938597794270, 18.86974299051346642288932904786, 19.65213805320562397170065268027, 20.04599203459854413175225342744, 20.71609420769857207861867161499, 21.65695550831897522590955037068

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