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Properties

Label 2-80-80.27-c1-0-5
Degree 22
Conductor 8080
Sign 0.454+0.890i0.454 + 0.890i
Analytic cond. 0.6388030.638803
Root an. cond. 0.7992510.799251
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(80s/2 ΓC(s) L(s)=((0.454+0.890i) Λ‾(2−s)\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}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.454+0.890i) Λ‾(1−s)\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: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.454+0.890i0.454 + 0.890i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(27,⋅)\chi_{80} (27, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), 0.454+0.890i)(2,\ 80,\ (\ :1/2),\ 0.454 + 0.890i)

Particular Values

L(1)L(1) ≈\approx 0.820985−0.502484i0.820985 - 0.502484i
L(12)L(\frac12) ≈\approx 0.820985−0.502484i0.820985 - 0.502484i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.516+1.31i)T 1 + (0.516 + 1.31i)T
5 1+(−2.07+0.841i)T 1 + (-2.07 + 0.841i)T
good3 1−1.28T+3T2 1 - 1.28T + 3T^{2}
7 1+(1.13+1.13i)T+7iT2 1 + (1.13 + 1.13i)T + 7iT^{2}
11 1+(2.32−2.32i)T−11iT2 1 + (2.32 - 2.32i)T - 11iT^{2}
13 1+1.36iT−13T2 1 + 1.36iT - 13T^{2}
17 1+(−5.25−5.25i)T+17iT2 1 + (-5.25 - 5.25i)T + 17iT^{2}
19 1+(3.69−3.69i)T−19iT2 1 + (3.69 - 3.69i)T - 19iT^{2}
23 1+(0.911−0.911i)T−23iT2 1 + (0.911 - 0.911i)T - 23iT^{2}
29 1+(2.37+2.37i)T+29iT2 1 + (2.37 + 2.37i)T + 29iT^{2}
31 1−0.242iT−31T2 1 - 0.242iT - 31T^{2}
37 1+3.34iT−37T2 1 + 3.34iT - 37T^{2}
41 1−2.66iT−41T2 1 - 2.66iT - 41T^{2}
43 1+9.04iT−43T2 1 + 9.04iT - 43T^{2}
47 1+(−7.87+7.87i)T−47iT2 1 + (-7.87 + 7.87i)T - 47iT^{2}
53 1+5.80T+53T2 1 + 5.80T + 53T^{2}
59 1+(−5.91−5.91i)T+59iT2 1 + (-5.91 - 5.91i)T + 59iT^{2}
61 1+(6.67−6.67i)T−61iT2 1 + (6.67 - 6.67i)T - 61iT^{2}
67 1+4.54iT−67T2 1 + 4.54iT - 67T^{2}
71 1−15.4T+71T2 1 - 15.4T + 71T^{2}
73 1+(1.49+1.49i)T+73iT2 1 + (1.49 + 1.49i)T + 73iT^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1−3.26T+83T2 1 - 3.26T + 83T^{2}
89 1−9.77T+89T2 1 - 9.77T + 89T^{2}
97 1+(1.63+1.63i)T+97iT2 1 + (1.63 + 1.63i)T + 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line