↑

Properties

Label 1-80-80.53-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

Related objects

Downloads

Learn more

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: $\chi_{80} (53, \cdot )$
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 \( 1 \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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