↑

Properties

Label 2-80-20.7-c1-0-1
Degree 22
Conductor 8080
Sign 0.880−0.473i0.880 - 0.473i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + 1.73i)3-s + (−1 − 2i)5-s + (−1.73 + 1.73i)7-s + 2.99i·9-s − 3.46i·11-s + (1 − i)13-s + (1.73 − 5.19i)15-s + (1 + i)17-s − 6.92·19-s − 5.99·21-s + (1.73 + 1.73i)23-s + (−3 + 4i)25-s − 4i·29-s + 3.46i·31-s + (5.99 − 5.99i)33-s + ⋯
L(s)  = 1  + (0.999 + 0.999i)3-s + (−0.447 − 0.894i)5-s + (−0.654 + 0.654i)7-s + 0.999i·9-s − 1.04i·11-s + (0.277 − 0.277i)13-s + (0.447 − 1.34i)15-s + (0.242 + 0.242i)17-s − 1.58·19-s − 1.30·21-s + (0.361 + 0.361i)23-s + (−0.600 + 0.800i)25-s − 0.742i·29-s + 0.622i·31-s + (1.04 − 1.04i)33-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((0.880−0.473i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.880−0.473i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.880−0.473i0.880 - 0.473i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(47,⋅)\chi_{80} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), 0.880−0.473i)(2,\ 80,\ (\ :1/2),\ 0.880 - 0.473i)

Particular Values

L(1)L(1) ≈\approx 1.07761+0.271502i1.07761 + 0.271502i
L(12)L(\frac12) ≈\approx 1.07761+0.271502i1.07761 + 0.271502i
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 1
5 1+(1+2i)T 1 + (1 + 2i)T
good3 1+(−1.73−1.73i)T+3iT2 1 + (-1.73 - 1.73i)T + 3iT^{2}
7 1+(1.73−1.73i)T−7iT2 1 + (1.73 - 1.73i)T - 7iT^{2}
11 1+3.46iT−11T2 1 + 3.46iT - 11T^{2}
13 1+(−1+i)T−13iT2 1 + (-1 + i)T - 13iT^{2}
17 1+(−1−i)T+17iT2 1 + (-1 - i)T + 17iT^{2}
19 1+6.92T+19T2 1 + 6.92T + 19T^{2}
23 1+(−1.73−1.73i)T+23iT2 1 + (-1.73 - 1.73i)T + 23iT^{2}
29 1+4iT−29T2 1 + 4iT - 29T^{2}
31 1−3.46iT−31T2 1 - 3.46iT - 31T^{2}
37 1+(−5−5i)T+37iT2 1 + (-5 - 5i)T + 37iT^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+(−1.73−1.73i)T+43iT2 1 + (-1.73 - 1.73i)T + 43iT^{2}
47 1+(1.73−1.73i)T−47iT2 1 + (1.73 - 1.73i)T - 47iT^{2}
53 1+(7−7i)T−53iT2 1 + (7 - 7i)T - 53iT^{2}
59 1−6.92T+59T2 1 - 6.92T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1+(−5.19+5.19i)T−67iT2 1 + (-5.19 + 5.19i)T - 67iT^{2}
71 1+10.3iT−71T2 1 + 10.3iT - 71T^{2}
73 1+(7−7i)T−73iT2 1 + (7 - 7i)T - 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1+(12.1+12.1i)T+83iT2 1 + (12.1 + 12.1i)T + 83iT^{2}
89 1+8iT−89T2 1 + 8iT - 89T^{2}
97 1+(7+7i)T+97iT2 1 + (7 + 7i)T + 97iT^{2}
show more
show less
   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

−14.71090846090380778188436539511, −13.42494735283822726129684489878, −12.50690298290117027560355194327, −11.08610604803245630663376174860, −9.751647727107471937945948420299, −8.798299079799299029076745321181, −8.206117683332553297486206698747, −5.97394988416631984998054114805, −4.35629388791130093372305338230, −3.11052228621699238958233508278, 2.41274550187004575203414873566, 3.93949304533825941930484225373, 6.65081477425201624883865452323, 7.24424605309752547319996364290, 8.391230144772941163596719172237, 9.825688468270321541522511683163, 11.01609878409626437252855464725, 12.52613665865153356054539447726, 13.20980434599533370792205502058, 14.38612350534080979559586096360

Graph of the ZZ-function along the critical line