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Properties

Label 1-1045-1045.1038-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.994 + 0.103i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.994 + 0.103i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.994 + 0.103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05344229079 - 1.025881209i\)
\(L(\frac12)\) \(\approx\) \(-0.05344229079 - 1.025881209i\)
\(L(1)\) \(\approx\) \(0.6023021570 - 0.6812995303i\)
\(L(1)\) \(\approx\) \(0.6023021570 - 0.6812995303i\)

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

−21.82271527105337643619132910911, −21.497054481757257028566713504626, −20.66375849120412387297790946102, −19.629541536037887717998415851427, −18.643708164926165257939560602512, −17.53893481425261457748688671499, −17.09022499021843020992425369105, −16.611366841009835177164865344231, −15.650831871108487856127811103930, −14.94284931688428931699064032077, −14.28224315288542723689283017108, −13.62918978434699117205113373605, −12.39563391055602391709394986492, −11.691092517528992698492978886387, −10.55351905573479556601842373912, −9.93529496154264170931223355989, −8.99630444619194070026751880660, −8.179745167826194397596161263514, −7.196497810281171760999370860833, −6.46069019478302247525235474149, −5.33236529721225643838972364601, −4.782388277321221745992152935808, −3.9918526714987558738968496915, −3.12509413279306281764132842250, −1.19943297732769408563487194538, 0.48261132190232212398780300987, 1.62659670908345829940551511700, 2.47857708217077393277603555501, 3.232594451611159901187959066294, 4.895953393588983844800179724196, 5.21700075993229691310708439032, 6.20824512723816761696799406331, 7.4285576993967584256875131432, 8.25593893332524348072838127094, 9.070458235109715002198640579775, 10.12805348470842268686902111170, 10.9944368942544680682580859584, 11.74881246965201485895591432591, 12.348014165500859370637153377295, 12.896922535667755525138530450191, 13.93032007176397393157839930676, 14.54383507689806665018168107070, 15.42806538429804619228519701418, 16.82828648794431771035849739798, 17.49006033672589259750155374327, 18.34104597359363864344305784657, 18.693551779603127723071620621565, 19.542750369492833447165461787535, 20.33020671627750153447336514118, 21.11627496320548131271500593229

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