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Properties

Label 1-80-80.77-r1-0-0
Degree $1$
Conductor $80$
Sign $0.811 - 0.584i$
Analytic cond. $8.59719$
Root an. cond. $8.59719$
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  + 3-s − i·7-s + 9-s − i·11-s + 13-s + i·17-s − i·19-s − i·21-s + i·23-s + 27-s − i·29-s + 31-s − i·33-s + 37-s + 39-s + ⋯
L(s)  = 1  + 3-s − i·7-s + 9-s − i·11-s + 13-s + i·17-s − i·19-s − i·21-s + i·23-s + 27-s − i·29-s + 31-s − i·33-s + 37-s + 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.811 - 0.584i$
Analytic conductor: \(8.59719\)
Root analytic conductor: \(8.59719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 80,\ (1:\ ),\ 0.811 - 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.271252239 - 0.7332120210i\)
\(L(\frac12)\) \(\approx\) \(2.271252239 - 0.7332120210i\)
\(L(1)\) \(\approx\) \(1.550513352 - 0.2516136789i\)
\(L(1)\) \(\approx\) \(1.550513352 - 0.2516136789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
23 \( 1 \)
29 \( 1 + T \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 + T \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−31.06369331349626417962772268311, −30.13815476349118751500870527512, −28.68184163686437944517320874403, −27.68066446835005444845709551315, −26.53616242002296177167135640666, −25.26424156645589453195574554330, −25.00179199224667022406062033012, −23.40458809376920289218912874025, −22.15630537087124685327364608787, −20.89245564044313353833743710069, −20.22310172907666407919762569683, −18.73565187395179511498527648981, −18.17816645060447605443286753300, −16.25350663056252687018835919917, −15.25501805260981540807383773290, −14.31060577420437244359348339543, −13.01475350714649859590284912902, −11.933123769522789876101780807300, −10.15850312839479714557795762006, −9.0355425789736495314653748171, −8.03201306169377786172639313051, −6.54670254616338078672835751927, −4.7771549248674510065261432672, −3.1762035813406245840757975159, −1.82248746391584786747254039559, 1.2224836850671518459759968478, 3.15545963381041502179461083914, 4.246156705039891246064325146, 6.31024301155343749023849894639, 7.74802142886863014152633508330, 8.73527029753967681993743820229, 10.11590443337437298504810513531, 11.29654037893721377191439233604, 13.296455455492518832818398384803, 13.67027127785175999856859248213, 15.08499197385339262967941615691, 16.18066719316313225643791831654, 17.51054576296235707316860990223, 18.96440826607694584299372494650, 19.75716776035155325493369210878, 20.855475871619430804493786418796, 21.75865812016353303004771435235, 23.43515639980748959383987521721, 24.21468313066071548421141619466, 25.55149588475102997456098321393, 26.37557986621399571332765811084, 27.19317049053399757561585667246, 28.59688354135926285925958461799, 30.08980420435824224533937667511, 30.44141257968821032580249511970

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