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Properties

Label 1-19e2-361.163-r0-0-0
Degree $1$
Conductor $361$
Sign $0.993 - 0.115i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (−0.754 − 0.656i)2-s + (−0.592 + 0.805i)3-s + (0.137 + 0.990i)4-s + (−0.962 + 0.272i)5-s + (0.975 − 0.218i)6-s + (0.789 + 0.614i)7-s + (0.546 − 0.837i)8-s + (−0.298 − 0.954i)9-s + (0.904 + 0.426i)10-s + (−0.677 − 0.735i)11-s + (−0.879 − 0.475i)12-s + (0.993 + 0.110i)13-s + (−0.191 − 0.981i)14-s + (0.350 − 0.936i)15-s + (−0.962 + 0.272i)16-s + (0.137 − 0.990i)17-s + ⋯
L(s)  = 1  + (−0.754 − 0.656i)2-s + (−0.592 + 0.805i)3-s + (0.137 + 0.990i)4-s + (−0.962 + 0.272i)5-s + (0.975 − 0.218i)6-s + (0.789 + 0.614i)7-s + (0.546 − 0.837i)8-s + (−0.298 − 0.954i)9-s + (0.904 + 0.426i)10-s + (−0.677 − 0.735i)11-s + (−0.879 − 0.475i)12-s + (0.993 + 0.110i)13-s + (−0.191 − 0.981i)14-s + (0.350 − 0.936i)15-s + (−0.962 + 0.272i)16-s + (0.137 − 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.993 - 0.115i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.993 - 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5952426370 - 0.03457268367i\)
\(L(\frac12)\) \(\approx\) \(0.5952426370 - 0.03457268367i\)
\(L(1)\) \(\approx\) \(0.5821154419 + 0.005771500786i\)
\(L(1)\) \(\approx\) \(0.5821154419 + 0.005771500786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.754 - 0.656i)T \)
3 \( 1 + (-0.592 + 0.805i)T \)
5 \( 1 + (-0.962 + 0.272i)T \)
7 \( 1 + (0.789 + 0.614i)T \)
11 \( 1 + (-0.677 - 0.735i)T \)
13 \( 1 + (0.993 + 0.110i)T \)
17 \( 1 + (0.137 - 0.990i)T \)
23 \( 1 + (-0.592 - 0.805i)T \)
29 \( 1 + (-0.926 + 0.376i)T \)
31 \( 1 + (0.945 + 0.324i)T \)
37 \( 1 + (-0.677 - 0.735i)T \)
41 \( 1 + (0.635 + 0.771i)T \)
43 \( 1 + (0.904 - 0.426i)T \)
47 \( 1 + (0.975 - 0.218i)T \)
53 \( 1 + (0.975 - 0.218i)T \)
59 \( 1 + (0.635 + 0.771i)T \)
61 \( 1 + (-0.998 + 0.0550i)T \)
67 \( 1 + (0.451 + 0.892i)T \)
71 \( 1 + (-0.998 - 0.0550i)T \)
73 \( 1 + (0.137 - 0.990i)T \)
79 \( 1 + (0.904 - 0.426i)T \)
83 \( 1 + (0.245 + 0.969i)T \)
89 \( 1 + (0.137 + 0.990i)T \)
97 \( 1 + (0.451 - 0.892i)T \)
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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

−24.44546057302590508593416830256, −23.95302721572895943268006625420, −23.321325416182266738816251394131, −22.72818143073819881828340996520, −20.89840192415856070993292060900, −20.09802678804730267305636100036, −19.18989352096198963378966993576, −18.45027063876942369184291021363, −17.543641962650369314403211679985, −17.00350173899211150104002403707, −15.89619604552095410398511452541, −15.223143250185637634654857158655, −14.00789167938213134699254057366, −13.04744822708421001292797153237, −11.85324178682116248900009731201, −11.0150913665133324408439663351, −10.31370702175249278949633625466, −8.67059468487354983263019106174, −7.7755222465014613169108111221, −7.48632470242345272380823379259, −6.20635633157006160620453486035, −5.19903917320616084473239177705, −4.10997246193790297618540567562, −1.93094563417554865132639948716, −0.89977560610716094463116762153, 0.74536107080570370010964670024, 2.61679448025103715500210651932, 3.64790791201515348670026287376, 4.61899337368629243199289280750, 5.88669278923133044499630807369, 7.32366727818867097223688262318, 8.4109818540013375086860702749, 8.99564367529108027176451164706, 10.4099435589260880310336661529, 11.03792655442407397477479199729, 11.65766255960438311112125701844, 12.42508145385626617621146081663, 13.95383056594333872780305636854, 15.2598087174479589024238906100, 16.019504641314860154528525591442, 16.54482594461676919751316121579, 18.00139493830572139902363663363, 18.32897289951662458718145352558, 19.30763720625174449666393972273, 20.68528689569828247314352701248, 20.865180072440351255889746859, 21.96429235842975024725072896800, 22.724444607415057058122799421590, 23.69774699778005810786083323407, 24.7332522657898231280100228190

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