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Properties

Label 1-4235-4235.3698-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.948 - 0.316i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.730 − 0.683i)2-s + (−0.743 + 0.669i)3-s + (0.0665 − 0.997i)4-s + (−0.0855 + 0.996i)6-s + (−0.633 − 0.774i)8-s + (0.104 − 0.994i)9-s + (0.618 + 0.786i)12-s + (0.884 + 0.466i)13-s + (−0.991 − 0.132i)16-s + (0.318 − 0.948i)17-s + (−0.603 − 0.797i)18-s + (−0.595 − 0.803i)19-s + (−0.690 + 0.723i)23-s + (0.988 + 0.151i)24-s + (0.964 − 0.263i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.730 − 0.683i)2-s + (−0.743 + 0.669i)3-s + (0.0665 − 0.997i)4-s + (−0.0855 + 0.996i)6-s + (−0.633 − 0.774i)8-s + (0.104 − 0.994i)9-s + (0.618 + 0.786i)12-s + (0.884 + 0.466i)13-s + (−0.991 − 0.132i)16-s + (0.318 − 0.948i)17-s + (−0.603 − 0.797i)18-s + (−0.595 − 0.803i)19-s + (−0.690 + 0.723i)23-s + (0.988 + 0.151i)24-s + (0.964 − 0.263i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.948 - 0.316i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3698, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.948 - 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1865293760 - 1.146869940i\)
\(L(\frac12)\) \(\approx\) \(0.1865293760 - 1.146869940i\)
\(L(1)\) \(\approx\) \(0.9966409140 - 0.4499525811i\)
\(L(1)\) \(\approx\) \(0.9966409140 - 0.4499525811i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.730 - 0.683i)T \)
3 \( 1 + (-0.743 + 0.669i)T \)
13 \( 1 + (0.884 + 0.466i)T \)
17 \( 1 + (0.318 - 0.948i)T \)
19 \( 1 + (-0.595 - 0.803i)T \)
23 \( 1 + (-0.690 + 0.723i)T \)
29 \( 1 + (-0.736 + 0.676i)T \)
31 \( 1 + (0.272 + 0.962i)T \)
37 \( 1 + (0.768 - 0.640i)T \)
41 \( 1 + (-0.516 - 0.856i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (0.603 - 0.797i)T \)
53 \( 1 + (-0.132 - 0.991i)T \)
59 \( 1 + (-0.483 - 0.875i)T \)
61 \( 1 + (0.290 + 0.956i)T \)
67 \( 1 + (-0.945 + 0.327i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (-0.0760 - 0.997i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (-0.336 - 0.941i)T \)
89 \( 1 + (0.995 - 0.0950i)T \)
97 \( 1 + (-0.931 + 0.362i)T \)
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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

−18.54097660305971499549551649500, −17.87806007631088707039179843330, −17.12341356869268524343772714653, −16.69727716312310408642863375058, −16.07514571213721799509317621198, −15.237680291278511699928422646037, −14.662711073019994671316270610195, −13.79295509675188062017842998526, −13.20645699908970181648820483082, −12.66212861296020052272165192457, −12.09703452064998291679715667847, −11.30062664605727431452131204150, −10.71502921990634169442403410823, −9.82285387142991782032645170564, −8.59504160587118824265940839024, −7.973006956775128409711742362649, −7.615939029616064792500157257138, −6.44806441278778603086580168115, −6.03849002740903048184784308577, −5.70978921191859796023712030063, −4.49501818488428084431989888992, −4.08701326366061945092045976174, −3.00637333384029571573791183304, −2.10799545070096048458519472275, −1.16334952532644520738526138898, 0.28886181248189178662797155897, 1.29417496744223389153992128411, 2.225253037021593401686686157917, 3.30423209055574262277231162711, 3.82144177042253323833099430353, 4.5895309083212295089378234479, 5.28161413743862160580698773689, 5.86443893900985124425050957559, 6.64562514805539936746504306716, 7.31731050855192194184149524721, 8.84265191756785435563098110156, 9.18308336545519689191834903013, 10.086456218309346059189434829753, 10.674885654383213454875503587317, 11.28407052710274854041689649776, 11.83340378272983360628264469625, 12.44140002465612529385859478111, 13.32195556665109736936677319284, 13.89533961168779830164123757468, 14.67131360913698920552913946621, 15.370803610646465230086343975494, 16.01233246766361066266122991940, 16.440212022037617014009502692448, 17.52618400217089748290398271434, 18.08999082610233374809211748092

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