Properties

Degree $1$
Conductor $80$
Sign $0.382 - 0.923i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

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Normalization:  

Dirichlet series

L(χ,s)  = 1  i·3-s + 7-s − 9-s i·11-s i·13-s − 17-s + i·19-s i·21-s + 23-s + i·27-s + i·29-s + 31-s − 33-s + i·37-s − 39-s + ⋯
L(s,χ)  = 1  i·3-s + 7-s − 9-s i·11-s i·13-s − 17-s + i·19-s i·21-s + 23-s + i·27-s + i·29-s + 31-s − 33-s + i·37-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.382 - 0.923i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.382 - 0.923i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.382 - 0.923i$
Motivic weight: \(0\)
Character: $\chi_{80} (29, \cdot )$
Sato-Tate group: $\mu(4)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 80,\ (0:\ ),\ 0.382 - 0.923i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.8260127328 - 0.5519240627i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.8260127328 - 0.5519240627i\)
\(L(\chi,1)\) \(\approx\) \(0.9839653104 - 0.3832669404i\)
\(L(1,\chi)\) \(\approx\) \(0.9839653104 - 0.3832669404i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.1819382442786483882311383340, −30.546622719007843407338031269010, −28.75498198510561096526677483866, −28.106589680856581857212138861226, −26.944200695211608974798993715620, −26.21444654009367201919358164095, −24.89612656046357304540125665145, −23.66103400298160350594195562661, −22.508867896907486379394383913050, −21.39102067166063803782165216995, −20.65203740813096546334298468384, −19.49169218233327118318772332866, −17.80740959593962514697640589568, −17.03267222227823698556855243781, −15.57597446879321769374456961594, −14.83079874057690524327453075322, −13.608206122097058135932588576995, −11.78324057926041452650479032396, −10.92467344002116528093198739649, −9.563256202155233061633146092583, −8.52666298079652404398795408122, −6.87615270492179283567050509004, −5.01760835254425728601443608627, −4.242702218139281208387819565778, −2.26556416171371388812607110302, 1.33985303017503609911230728961, 3.01958772020434558372035280211, 5.112395139536241438446545903338, 6.40889075112605285372674048086, 7.87394369825586160490294363247, 8.64591087195088022788642137309, 10.71799306365399963466266115915, 11.69085896407754923210233545867, 12.98760202279609519162522342768, 13.98821298669171548038780284670, 15.12735637977140731906458604941, 16.80527558071635644037083227062, 17.8326765320838754041947939592, 18.696478831944082762066794955746, 19.89716872303525536055565047553, 20.97620481417272496321555878610, 22.37556361328482793974270645193, 23.542161452659345053982640867818, 24.49750043521466543640798745569, 25.15260915442748760015077800389, 26.67820309359247613239574096845, 27.65089463596872782844600423332, 29.01954499842091074750527549247, 29.80663786430008132136412130178, 30.817728631060320917065773602575

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