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Properties

Label 2-80-80.69-c1-0-4
Degree 22
Conductor 8080
Sign 0.755−0.655i0.755 - 0.655i
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  + (1 + i)2-s + (−1 − i)3-s + 2i·4-s + (2 + i)5-s − 2i·6-s + (−2 + 2i)8-s − i·9-s + (1 + 3i)10-s + (−3 − 3i)11-s + (2 − 2i)12-s + (−3 − 3i)13-s + (−1 − 3i)15-s − 4·16-s + 4i·17-s + (1 − i)18-s + (−1 + i)19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.577 − 0.577i)3-s + i·4-s + (0.894 + 0.447i)5-s − 0.816i·6-s + (−0.707 + 0.707i)8-s − 0.333i·9-s + (0.316 + 0.948i)10-s + (−0.904 − 0.904i)11-s + (0.577 − 0.577i)12-s + (−0.832 − 0.832i)13-s + (−0.258 − 0.774i)15-s − 16-s + 0.970i·17-s + (0.235 − 0.235i)18-s + (−0.229 + 0.229i)19-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((0.755−0.655i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.755−0.655i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.755−0.655i0.755 - 0.655i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(69,⋅)\chi_{80} (69, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), 0.755−0.655i)(2,\ 80,\ (\ :1/2),\ 0.755 - 0.655i)

Particular Values

L(1)L(1) ≈\approx 1.10789+0.413506i1.10789 + 0.413506i
L(12)L(\frac12) ≈\approx 1.10789+0.413506i1.10789 + 0.413506i
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−i)T 1 + (-1 - i)T
5 1+(−2−i)T 1 + (-2 - i)T
good3 1+(1+i)T+3iT2 1 + (1 + i)T + 3iT^{2}
7 1+7T2 1 + 7T^{2}
11 1+(3+3i)T+11iT2 1 + (3 + 3i)T + 11iT^{2}
13 1+(3+3i)T+13iT2 1 + (3 + 3i)T + 13iT^{2}
17 1−4iT−17T2 1 - 4iT - 17T^{2}
19 1+(1−i)T−19iT2 1 + (1 - i)T - 19iT^{2}
23 1−8T+23T2 1 - 8T + 23T^{2}
29 1+(−3+3i)T−29iT2 1 + (-3 + 3i)T - 29iT^{2}
31 1+31T2 1 + 31T^{2}
37 1+(3−3i)T−37iT2 1 + (3 - 3i)T - 37iT^{2}
41 1−41T2 1 - 41T^{2}
43 1+(3−3i)T−43iT2 1 + (3 - 3i)T - 43iT^{2}
47 1−2iT−47T2 1 - 2iT - 47T^{2}
53 1+(−9+9i)T−53iT2 1 + (-9 + 9i)T - 53iT^{2}
59 1+(−9−9i)T+59iT2 1 + (-9 - 9i)T + 59iT^{2}
61 1+(5−5i)T−61iT2 1 + (5 - 5i)T - 61iT^{2}
67 1+(−3−3i)T+67iT2 1 + (-3 - 3i)T + 67iT^{2}
71 1−6iT−71T2 1 - 6iT - 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+(9+9i)T+83iT2 1 + (9 + 9i)T + 83iT^{2}
89 1+12iT−89T2 1 + 12iT - 89T^{2}
97 1+12iT−97T2 1 + 12iT - 97T^{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

−14.57742198816264828588736620394, −13.21263327872961666426845699339, −12.85605969118348066594036764241, −11.49524970232935581612553876823, −10.26075897927273957609018588390, −8.550786391460375997190461702687, −7.18837648543051429883700073371, −6.13044312346254729524365106867, −5.28560249338780965286128836829, −3.00055124653600007170672131703, 2.35800388573391443252054775612, 4.81303490109649790741175582599, 5.16957075528633276263265883990, 6.91565115638612820206505696023, 9.228785443790332110863380008958, 10.07059388567639789516193372782, 10.96056544901509384935604857787, 12.17801621362332164398493210095, 13.12580103815786168627567314299, 14.04901381589848517669287105152

Graph of the ZZ-function along the critical line