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Properties

Label 1-80-80.69-r0-0-0
Degree $1$
Conductor $80$
Sign $0.382 + 0.923i$
Analytic cond. $0.371518$
Root an. cond. $0.371518$
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  + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(0.371518\)
Root analytic conductor: \(0.371518\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 80,\ (0:\ ),\ 0.382 + 0.923i)\)

Particular Values

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

Euler product

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

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

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