Properties

Label 1-1045-1045.512-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.544 + 0.838i$
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  + (−0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.809 + 0.587i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s i·12-s + (0.587 + 0.809i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.587 + 0.809i)17-s + (−0.951 − 0.309i)18-s + 21-s i·23-s + (−0.809 + 0.587i)24-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.809 + 0.587i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s i·12-s + (0.587 + 0.809i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.587 + 0.809i)17-s + (−0.951 − 0.309i)18-s + 21-s i·23-s + (−0.809 + 0.587i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.544 + 0.838i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.544 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1226185427 + 0.2257279690i\)
\(L(\frac12)\) \(\approx\) \(0.1226185427 + 0.2257279690i\)
\(L(1)\) \(\approx\) \(0.4862254224 - 0.07375389932i\)
\(L(1)\) \(\approx\) \(0.4862254224 - 0.07375389932i\)

Euler product

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

−21.15031537304140331516267820432, −20.01288100768512874963491514870, −19.185564255295233080855535576724, −18.63020058956392065400583938287, −17.77570750108881664300442587205, −17.30961638465160024830559352242, −16.36779585603965595838828494080, −15.642954889205494838104175385261, −15.391654164799337382308039711585, −13.76793582080314088959667733813, −13.36555668707158778739769388599, −12.33674698633790698413294481992, −11.41265587148878283994038800130, −10.52472406540846413152236010605, −9.83794500995280997551143673814, −8.98321857896582943350208633719, −7.96615078555241364378015633113, −7.06589496965679790215028775626, −6.43059085150324020530252527296, −5.64673173962409654408981708587, −5.00160048428483122588964653764, −3.69746519563111805351110136005, −2.22415896434845402167408978699, −0.934720866291120434912224038, −0.11639677325694284093527905690, 0.89252777307394424159723291155, 1.97581935955858705486047413785, 3.32900513575219929989775780747, 4.05215944577211154564973058338, 4.88610663013925524399995205071, 6.40626988277011107443508330683, 6.681348417209612831375748865834, 8.01493195611226014038368029058, 9.052450820715767322901830684760, 9.713685338821910498957157418420, 10.54052919037726687024595123633, 11.092546834085391790947789220609, 11.9253761979458329422801040024, 12.81098106277994340583695248958, 13.22468867642739319126767250020, 14.492902580034614810450717312284, 15.8318193928795804666510021034, 16.30940683928436728793279340875, 16.99006187173012830919328033125, 17.73355770229518212743842091970, 18.562597105217925977755366048516, 19.192234042472055878635453087883, 20.01467086406803205673655184755, 20.93345143320904867293403920521, 21.5264009969435869434394884952

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