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Properties

Label 2-80-80.3-c1-0-2
Degree $2$
Conductor $80$
Sign $0.454 - 0.890i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.516 + 1.31i)2-s + 1.28·3-s + (−1.46 − 1.36i)4-s + (2.07 + 0.841i)5-s + (−0.662 + 1.68i)6-s + (−1.13 + 1.13i)7-s + (2.54 − 1.22i)8-s − 1.35·9-s + (−2.17 + 2.29i)10-s + (−2.32 − 2.32i)11-s + (−1.87 − 1.74i)12-s + 1.36i·13-s + (−0.911 − 2.08i)14-s + (2.65 + 1.07i)15-s + (0.297 + 3.98i)16-s + (5.25 − 5.25i)17-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)2-s + 0.739·3-s + (−0.732 − 0.680i)4-s + (0.926 + 0.376i)5-s + (−0.270 + 0.688i)6-s + (−0.430 + 0.430i)7-s + (0.901 − 0.433i)8-s − 0.452·9-s + (−0.688 + 0.724i)10-s + (−0.700 − 0.700i)11-s + (−0.542 − 0.503i)12-s + 0.378i·13-s + (−0.243 − 0.558i)14-s + (0.685 + 0.278i)15-s + (0.0744 + 0.997i)16-s + (1.27 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820985 + 0.502484i\)
\(L(\frac12)\) \(\approx\) \(0.820985 + 0.502484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.516 - 1.31i)T \)
5 \( 1 + (-2.07 - 0.841i)T \)
good3 \( 1 - 1.28T + 3T^{2} \)
7 \( 1 + (1.13 - 1.13i)T - 7iT^{2} \)
11 \( 1 + (2.32 + 2.32i)T + 11iT^{2} \)
13 \( 1 - 1.36iT - 13T^{2} \)
17 \( 1 + (-5.25 + 5.25i)T - 17iT^{2} \)
19 \( 1 + (3.69 + 3.69i)T + 19iT^{2} \)
23 \( 1 + (0.911 + 0.911i)T + 23iT^{2} \)
29 \( 1 + (2.37 - 2.37i)T - 29iT^{2} \)
31 \( 1 + 0.242iT - 31T^{2} \)
37 \( 1 - 3.34iT - 37T^{2} \)
41 \( 1 + 2.66iT - 41T^{2} \)
43 \( 1 - 9.04iT - 43T^{2} \)
47 \( 1 + (-7.87 - 7.87i)T + 47iT^{2} \)
53 \( 1 + 5.80T + 53T^{2} \)
59 \( 1 + (-5.91 + 5.91i)T - 59iT^{2} \)
61 \( 1 + (6.67 + 6.67i)T + 61iT^{2} \)
67 \( 1 - 4.54iT - 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (1.49 - 1.49i)T - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 3.26T + 83T^{2} \)
89 \( 1 - 9.77T + 89T^{2} \)
97 \( 1 + (1.63 - 1.63i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.43007696586270425684229907165, −13.94591109732664185281184535238, −12.94128656867917604875058299039, −10.99350542815591883765181350326, −9.662962407232584177181099776183, −8.987738202986269881935135407286, −7.77983732201985659571703268865, −6.37219222552418291322396045844, −5.30725364707375854718797864753, −2.83849966474416559970015879626, 2.12465614339622644752936543920, 3.71502546718775458421505513085, 5.62371777204037166138815658596, 7.77544458698373091190194800614, 8.734604720205434827835677184253, 9.955346989638666686820910924920, 10.46946280628922074534058066252, 12.29389435586936793981524525252, 13.03696507859897265674346255587, 13.92296499365289449809594290116

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