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Properties

Label 1-389-389.80-r0-0-0
Degree $1$
Conductor $389$
Sign $-0.985 - 0.169i$
Analytic cond. $1.80650$
Root an. cond. $1.80650$
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  + (0.145 + 0.989i)2-s + (−0.974 + 0.224i)3-s + (−0.957 + 0.287i)4-s + (0.948 + 0.318i)5-s + (−0.363 − 0.931i)6-s + (−0.777 + 0.628i)7-s + (−0.423 − 0.905i)8-s + (0.898 − 0.438i)9-s + (−0.177 + 0.984i)10-s + (0.997 + 0.0647i)11-s + (0.868 − 0.495i)12-s + (−0.0485 + 0.998i)13-s + (−0.735 − 0.677i)14-s + (−0.995 − 0.0970i)15-s + (0.834 − 0.550i)16-s + (−0.177 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.145 + 0.989i)2-s + (−0.974 + 0.224i)3-s + (−0.957 + 0.287i)4-s + (0.948 + 0.318i)5-s + (−0.363 − 0.931i)6-s + (−0.777 + 0.628i)7-s + (−0.423 − 0.905i)8-s + (0.898 − 0.438i)9-s + (−0.177 + 0.984i)10-s + (0.997 + 0.0647i)11-s + (0.868 − 0.495i)12-s + (−0.0485 + 0.998i)13-s + (−0.735 − 0.677i)14-s + (−0.995 − 0.0970i)15-s + (0.834 − 0.550i)16-s + (−0.177 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ -0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06982598248 + 0.8179607647i\)
\(L(\frac12)\) \(\approx\) \(-0.06982598248 + 0.8179607647i\)
\(L(1)\) \(\approx\) \(0.5220717979 + 0.6193726825i\)
\(L(1)\) \(\approx\) \(0.5220717979 + 0.6193726825i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.145 + 0.989i)T \)
3 \( 1 + (-0.974 + 0.224i)T \)
5 \( 1 + (0.948 + 0.318i)T \)
7 \( 1 + (-0.777 + 0.628i)T \)
11 \( 1 + (0.997 + 0.0647i)T \)
13 \( 1 + (-0.0485 + 0.998i)T \)
17 \( 1 + (-0.177 + 0.984i)T \)
19 \( 1 + (-0.884 + 0.466i)T \)
23 \( 1 + (0.948 - 0.318i)T \)
29 \( 1 + (-0.536 - 0.843i)T \)
31 \( 1 + (-0.641 + 0.767i)T \)
37 \( 1 + (0.616 + 0.787i)T \)
41 \( 1 + (-0.884 - 0.466i)T \)
43 \( 1 + (-0.302 + 0.953i)T \)
47 \( 1 + (-0.884 - 0.466i)T \)
53 \( 1 + (0.0808 - 0.996i)T \)
59 \( 1 + (0.0161 - 0.999i)T \)
61 \( 1 + (-0.0485 + 0.998i)T \)
67 \( 1 + (-0.912 - 0.408i)T \)
71 \( 1 + (-0.957 + 0.287i)T \)
73 \( 1 + (0.834 + 0.550i)T \)
79 \( 1 + (-0.641 - 0.767i)T \)
83 \( 1 + (-0.912 + 0.408i)T \)
89 \( 1 + (0.991 + 0.129i)T \)
97 \( 1 + (-0.957 - 0.287i)T \)
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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

−23.72021930453919443292313980535, −22.8494099026123294989666534323, −22.255835881601471929159146756159, −21.6115175320750396653721563361, −20.49199950667202818317118785981, −19.78199800363504503629460396393, −18.74661418897046611338030515585, −17.86706551584320014027440164396, −17.124520425728350760778563824112, −16.52985775742008047677320325636, −14.98663776351761533278897506861, −13.71423683672171405926139018085, −13.07009832591088515491395812443, −12.49977070832816234422769747909, −11.30555387893193452101587223657, −10.57886758705307618265859867485, −9.696812880143738141619622709, −8.97698764323034917294356017559, −7.20592373702066101284018174316, −6.16637610946853873366688898692, −5.28072762483065462889253556224, −4.29774561683660828261227201533, −2.99666602730787689302252662186, −1.61000393518461053407390790110, −0.556219258821516503605195336470, 1.68161043043413030289982091096, 3.54813902662789926868546061476, 4.62425747156469290193386141023, 5.75070643489500185606939119786, 6.46936161404855118887964623114, 6.8183262115195594815927159027, 8.686607129868907772873829068723, 9.46209510614034176182940704344, 10.22603829503594959075284337989, 11.56585477673266482996951624494, 12.65001518753489356655661514698, 13.27623585112931205566357277785, 14.616703484409571594586890897410, 15.112693115496821868790996803566, 16.458091785043104113349748474386, 16.83719744517458252118377002028, 17.600613853866325388557984470375, 18.614010423184149996689935689563, 19.20978704276992759996273546894, 21.22856208336258151781361389075, 21.72700282287656166865284044256, 22.40024414732284903473569986869, 23.09640769582332141332274077328, 24.09278549631793946473307052947, 24.95254311674360497593193113270

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