Properties

Label 1-1045-1045.1029-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.659 + 0.751i$
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  + (0.719 + 0.694i)2-s + (0.848 + 0.529i)3-s + (0.0348 + 0.999i)4-s + (0.241 + 0.970i)6-s + (0.669 + 0.743i)7-s + (−0.669 + 0.743i)8-s + (0.438 + 0.898i)9-s + (−0.5 + 0.866i)12-s + (0.997 + 0.0697i)13-s + (−0.0348 + 0.999i)14-s + (−0.997 + 0.0697i)16-s + (0.438 − 0.898i)17-s + (−0.309 + 0.951i)18-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (−0.961 + 0.275i)24-s + ⋯
L(s)  = 1  + (0.719 + 0.694i)2-s + (0.848 + 0.529i)3-s + (0.0348 + 0.999i)4-s + (0.241 + 0.970i)6-s + (0.669 + 0.743i)7-s + (−0.669 + 0.743i)8-s + (0.438 + 0.898i)9-s + (−0.5 + 0.866i)12-s + (0.997 + 0.0697i)13-s + (−0.0348 + 0.999i)14-s + (−0.997 + 0.0697i)16-s + (0.438 − 0.898i)17-s + (−0.309 + 0.951i)18-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (−0.961 + 0.275i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.659 + 0.751i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.659 + 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.405179527 + 3.104690209i\)
\(L(\frac12)\) \(\approx\) \(1.405179527 + 3.104690209i\)
\(L(1)\) \(\approx\) \(1.607910237 + 1.487959705i\)
\(L(1)\) \(\approx\) \(1.607910237 + 1.487959705i\)

Euler product

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

−21.05736273216835889056521952304, −20.52096351786465860870502538445, −19.927730973096241786114714281417, −19.03142188191256563181724327072, −18.49365442384283897197278575147, −17.592788599169565947722431266372, −16.52817484768219877769989629865, −15.22022360579959973083605113696, −14.855454798016204982107798390076, −13.97855341343550650090195734608, −13.3079636329835662983412345439, −12.84635635586730605914140704614, −11.73292819119027677476657301323, −11.01336497195919539721642541135, −10.19147825803900490801346169014, −9.23024602178834779917917723652, −8.33149341096326997332464555186, −7.44902031704891114807536931191, −6.49819671788726074074191240644, −5.57940249056581693568412737461, −4.3524232939976987861695954783, −3.69967567593784460706380846784, −2.86491287473116002548794178773, −1.62463499828740296103096130284, −1.14575324662731554175342208465, 1.75617737225003429333929454329, 2.834738708510342999127888328285, 3.5244830933610905068021621128, 4.61356606617425353541841447268, 5.19804205157768755370082206925, 6.18168290463535836033607118668, 7.3281466003072807523664806287, 8.06095708275697604012755814184, 8.819568715234210686347188058091, 9.40718456562510263737571965812, 10.83386410217751528969982935922, 11.52578364932634993796721070688, 12.56631681916805105013867699933, 13.39336288155057285793785410126, 14.12087805343392243100240103572, 14.78277361350798399174625744078, 15.397384534678449469018676152592, 16.13678660415042906946105297867, 16.771378263379999184407801989713, 18.025135753366288205992940565594, 18.523191386680765956820194489583, 19.66088583375455007679316966964, 20.74213689470840378847966305860, 21.05928637533719299741730672601, 21.673756936921249316793676191711

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