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Properties

Label 1-1045-1045.1039-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.490 + 0.871i$
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  + (0.438 + 0.898i)2-s + (−0.374 + 0.927i)3-s + (−0.615 + 0.788i)4-s + (−0.997 + 0.0697i)6-s + (0.978 − 0.207i)7-s + (−0.978 − 0.207i)8-s + (−0.719 − 0.694i)9-s + (−0.5 − 0.866i)12-s + (−0.241 + 0.970i)13-s + (0.615 + 0.788i)14-s + (−0.241 − 0.970i)16-s + (0.719 − 0.694i)17-s + (0.309 − 0.951i)18-s + (−0.173 + 0.984i)21-s + (0.939 + 0.342i)23-s + (0.559 − 0.829i)24-s + ⋯
L(s)  = 1  + (0.438 + 0.898i)2-s + (−0.374 + 0.927i)3-s + (−0.615 + 0.788i)4-s + (−0.997 + 0.0697i)6-s + (0.978 − 0.207i)7-s + (−0.978 − 0.207i)8-s + (−0.719 − 0.694i)9-s + (−0.5 − 0.866i)12-s + (−0.241 + 0.970i)13-s + (0.615 + 0.788i)14-s + (−0.241 − 0.970i)16-s + (0.719 − 0.694i)17-s + (0.309 − 0.951i)18-s + (−0.173 + 0.984i)21-s + (0.939 + 0.342i)23-s + (0.559 − 0.829i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.490 + 0.871i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.490 + 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.192148416 + 2.037698886i\)
\(L(\frac12)\) \(\approx\) \(1.192148416 + 2.037698886i\)
\(L(1)\) \(\approx\) \(0.8957804923 + 0.8519249112i\)
\(L(1)\) \(\approx\) \(0.8957804923 + 0.8519249112i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.438 + 0.898i)T \)
3 \( 1 + (-0.374 + 0.927i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.241 + 0.970i)T \)
17 \( 1 + (0.719 - 0.694i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.990 - 0.139i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.374 - 0.927i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.882 - 0.469i)T \)
53 \( 1 + (0.961 - 0.275i)T \)
59 \( 1 + (0.882 + 0.469i)T \)
61 \( 1 + (0.559 + 0.829i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.961 - 0.275i)T \)
73 \( 1 + (-0.848 + 0.529i)T \)
79 \( 1 + (0.997 + 0.0697i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.438 + 0.898i)T \)
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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

−20.99894036206844089101113591213, −20.35971812351163164974274444539, −19.52820928976206032629194735561, −18.77421212366569722569712014909, −18.151132621084862766774755808778, −17.482841716331162509187460130306, −16.69587383403209119081003273778, −15.16113032641235414166625371185, −14.639154806145189532952544195170, −13.84750227760270124484284188091, −12.758580158722185488448144651932, −12.60765164373113188151462349814, −11.49586649396575722128277285967, −11.000515693390244979172642032484, −10.20270665269629618864492988961, −8.96497684762240997950652634096, −8.132519746500253144342140481942, −7.32329857477850810164508145371, −6.00862847708473208811323917590, −5.42428451539132397933881162138, −4.63243869198005791578767781680, −3.327089611522260031736188119710, −2.39965684927812736029947789499, −1.47537323387803871375417947840, −0.71896729535413616456908214558, 0.68879599516942418872330185861, 2.421608109962973262347025937288, 3.78109897456923723092429206987, 4.2617299492253339697003317577, 5.32690976618240448561281258378, 5.63546829118761961446786353359, 7.05232990764350276046045916757, 7.56570606865060574711264954141, 8.89205101616619048949133066754, 9.226010448509043000288763520145, 10.41695265022946217897371033636, 11.44698036847266839396594642798, 11.900647620498291643367006458340, 13.07139852129379025754091663653, 14.141820818800973842625558217735, 14.57441279035327479424616902086, 15.26084199898374969249290339022, 16.23250680681608817558925734785, 16.73397147411444102467574943611, 17.41901207930019152445742033747, 18.16330248766440807380481516794, 19.12495306172717913267081414335, 20.6084183733102087689074060327, 20.929470002517289183410234229724, 21.710849691108964063468809938012

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