↑

Properties

Label 1-80-80.67-r0-0-0
Degree $1$
Conductor $80$
Sign $0.584 + 0.811i$
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

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

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

Invariants

Degree: \(1\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(0.371518\)
Root analytic conductor: \(0.371518\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 80,\ (0:\ ),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6339682297 + 0.3245399345i\)
\(L(\frac12)\) \(\approx\) \(0.6339682297 + 0.3245399345i\)
\(L(1)\) \(\approx\) \(0.7778306810 + 0.1779870287i\)
\(L(1)\) \(\approx\) \(0.7778306810 + 0.1779870287i\)

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

−30.64690726255690319434393684581, −29.65127699816237123076158782645, −28.96516734073685241832834054023, −27.6427855404113613333466260790, −26.91285085236944550979032672325, −25.61640061398727013582212482423, −24.0410968907796542563874581671, −23.50493037675563899585313148897, −22.347041692260585677251865149247, −21.311912084696456071037789681556, −20.06713713016063243652807572341, −18.66330652740909227684767165822, −17.662360358098822772934415319311, −16.539301139232663076596844062427, −15.789840615171618608915212304664, −13.92062402446148580480834221592, −13.022418824106849040642453520274, −11.360176363701513459222207756803, −10.83479787720429905238102849720, −9.305029601012511855866400981932, −7.53225598946094748022119420049, −6.368100549695411670306845243991, −5.041062915886929408690539307786, −3.574395538397809954876429989252, −1.00682263008344580235730018072, 1.856034201276730400315121349290, 4.09038026950480294906157572017, 5.56127913337550286982484363919, 6.51032361339326934190164254047, 8.183074427670075188474465574578, 9.725389790261117112437958790541, 10.92612016354067634293088089877, 12.14600011443515816830391833617, 12.91556385482205817356325057246, 14.80423116205896681836480141714, 15.80381188980294992828504895087, 16.95311108987040473183573972812, 18.12473380540240408814746160103, 18.84096265837214421912302005486, 20.576045914081226254771845481418, 21.64559905079838317165288346450, 22.655280040482929166789598822981, 23.522647976976510835974627217762, 24.7591279969111744552574603351, 25.77254331492708819128615079874, 27.267502533213404288487754130961, 28.31024844225530605480175109397, 28.70189021418782409970824201106, 30.220047191823563092358851250447, 31.02939268393739388910425054353

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