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Properties

Label 1-1045-1045.478-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.977 + 0.213i$
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  + (0.694 − 0.719i)2-s + (0.529 − 0.848i)3-s + (−0.0348 − 0.999i)4-s + (−0.241 − 0.970i)6-s + (0.743 − 0.669i)7-s + (−0.743 − 0.669i)8-s + (−0.438 − 0.898i)9-s + (−0.866 − 0.5i)12-s + (−0.0697 + 0.997i)13-s + (0.0348 − 0.999i)14-s + (−0.997 + 0.0697i)16-s + (−0.898 − 0.438i)17-s + (−0.951 − 0.309i)18-s + (−0.173 − 0.984i)21-s + (−0.342 − 0.939i)23-s + (−0.961 + 0.275i)24-s + ⋯
L(s)  = 1  + (0.694 − 0.719i)2-s + (0.529 − 0.848i)3-s + (−0.0348 − 0.999i)4-s + (−0.241 − 0.970i)6-s + (0.743 − 0.669i)7-s + (−0.743 − 0.669i)8-s + (−0.438 − 0.898i)9-s + (−0.866 − 0.5i)12-s + (−0.0697 + 0.997i)13-s + (0.0348 − 0.999i)14-s + (−0.997 + 0.0697i)16-s + (−0.898 − 0.438i)17-s + (−0.951 − 0.309i)18-s + (−0.173 − 0.984i)21-s + (−0.342 − 0.939i)23-s + (−0.961 + 0.275i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.977 + 0.213i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.977 + 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2456091352 - 2.279038550i\)
\(L(\frac12)\) \(\approx\) \(-0.2456091352 - 2.279038550i\)
\(L(1)\) \(\approx\) \(0.9767802885 - 1.355132961i\)
\(L(1)\) \(\approx\) \(0.9767802885 - 1.355132961i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.694 - 0.719i)T \)
3 \( 1 + (0.529 - 0.848i)T \)
7 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (-0.0697 + 0.997i)T \)
17 \( 1 + (-0.898 - 0.438i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (-0.882 - 0.469i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.848 - 0.529i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.139 - 0.990i)T \)
53 \( 1 + (-0.829 - 0.559i)T \)
59 \( 1 + (0.990 + 0.139i)T \)
61 \( 1 + (0.961 + 0.275i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (-0.559 - 0.829i)T \)
73 \( 1 + (0.927 - 0.374i)T \)
79 \( 1 + (-0.241 + 0.970i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.694 - 0.719i)T \)
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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

−21.97379832249331380041038267605, −21.414352522335149235477150572559, −20.515615094594312328176315001971, −20.03826554538481747404758478521, −18.78597008621661814121369918230, −17.72541503373464449889993571050, −17.239465920449154120586939275, −16.12762303403260358486312352482, −15.54209273501243544652510035171, −14.88884202490163912848765790884, −14.44119948948106948979388233394, −13.34993720865933311746199841447, −12.79311789623482854797695910220, −11.51270091848407677174526430911, −11.037692096127918940763744957896, −9.72646517182357839533893117883, −8.92420310808768595428806690190, −8.08236149858014299426521834009, −7.62299856934734106198268571563, −6.15655446956733896902685833341, −5.435840839497541168549699002151, −4.68945800405385120701031371569, −3.84873237895144567095486658469, −2.905532563252523376308843314665, −2.02755318936661333432645752902, 0.661235556919543818520953269786, 1.83606388009680934723963170115, 2.33502095635034323326865423419, 3.617961588523789330870241395716, 4.34439374662868910807011437442, 5.30922009630290826691206940599, 6.59052164995439054169912681284, 7.01035931987077525205469147003, 8.245101629698466308838018650619, 9.04044271396114612948153637918, 9.99484858763546702633588558498, 11.05837793688022221087824120814, 11.64436105263162231210171495062, 12.42538836668888010353425244953, 13.37430881206830160354427476278, 13.858699756031770640733041387471, 14.50935814276466822794563926238, 15.2177245120239844143184182881, 16.41798802053842357074696309294, 17.49896683635748474865755067544, 18.27212678589866920917563188230, 18.92082131681581420652495773929, 19.73644199992848311235024448517, 20.42353627687512934087046750518, 20.86260769147982637790358420317

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