Properties

Degree 1
Conductor $ 2^{4} \cdot 5 $
Sign $0.811 - 0.584i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $0.811 - 0.584i$
motivic weight  =  \(0\)
character  :  $\chi_{80} (77, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 80,\ (1:\ ),\ 0.811 - 0.584i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.271252239 - 0.7332120210i$
$L(\frac12,\chi)$  $\approx$  $2.271252239 - 0.7332120210i$
$L(\chi,1)$  $\approx$  1.550513352 - 0.2516136789i
$L(1,\chi)$  $\approx$  1.550513352 - 0.2516136789i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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