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Properties

Label 2-80-80.27-c1-0-6
Degree 22
Conductor 8080
Sign 0.994+0.109i0.994 + 0.109i
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  + (1.41 − 0.0660i)2-s − 0.496·3-s + (1.99 − 0.186i)4-s + (−2.00 − 0.987i)5-s + (−0.701 + 0.0328i)6-s + (1.55 + 1.55i)7-s + (2.80 − 0.395i)8-s − 2.75·9-s + (−2.89 − 1.26i)10-s + (−4.19 + 4.19i)11-s + (−0.988 + 0.0927i)12-s − 5.09i·13-s + (2.29 + 2.09i)14-s + (0.996 + 0.490i)15-s + (3.93 − 0.743i)16-s + (0.213 + 0.213i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0467i)2-s − 0.286·3-s + (0.995 − 0.0933i)4-s + (−0.897 − 0.441i)5-s + (−0.286 + 0.0133i)6-s + (0.587 + 0.587i)7-s + (0.990 − 0.139i)8-s − 0.917·9-s + (−0.916 − 0.399i)10-s + (−1.26 + 1.26i)11-s + (−0.285 + 0.0267i)12-s − 1.41i·13-s + (0.614 + 0.559i)14-s + (0.257 + 0.126i)15-s + (0.982 − 0.185i)16-s + (0.0517 + 0.0517i)17-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((0.994+0.109i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.994+0.109i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.994+0.109i0.994 + 0.109i
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), 0.994+0.109i)(2,\ 80,\ (\ :1/2),\ 0.994 + 0.109i)

Particular Values

L(1)L(1) ≈\approx 1.33117−0.0728634i1.33117 - 0.0728634i
L(12)L(\frac12) ≈\approx 1.33117−0.0728634i1.33117 - 0.0728634i
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+(−1.41+0.0660i)T 1 + (-1.41 + 0.0660i)T
5 1+(2.00+0.987i)T 1 + (2.00 + 0.987i)T
good3 1+0.496T+3T2 1 + 0.496T + 3T^{2}
7 1+(−1.55−1.55i)T+7iT2 1 + (-1.55 - 1.55i)T + 7iT^{2}
11 1+(4.19−4.19i)T−11iT2 1 + (4.19 - 4.19i)T - 11iT^{2}
13 1+5.09iT−13T2 1 + 5.09iT - 13T^{2}
17 1+(−0.213−0.213i)T+17iT2 1 + (-0.213 - 0.213i)T + 17iT^{2}
19 1+(−0.844+0.844i)T−19iT2 1 + (-0.844 + 0.844i)T - 19iT^{2}
23 1+(−1.70+1.70i)T−23iT2 1 + (-1.70 + 1.70i)T - 23iT^{2}
29 1+(−2.24−2.24i)T+29iT2 1 + (-2.24 - 2.24i)T + 29iT^{2}
31 1−0.818iT−31T2 1 - 0.818iT - 31T^{2}
37 1+5.12iT−37T2 1 + 5.12iT - 37T^{2}
41 1−3.34iT−41T2 1 - 3.34iT - 41T^{2}
43 1−4.49iT−43T2 1 - 4.49iT - 43T^{2}
47 1+(−4.29+4.29i)T−47iT2 1 + (-4.29 + 4.29i)T - 47iT^{2}
53 1+1.00T+53T2 1 + 1.00T + 53T^{2}
59 1+(7.65+7.65i)T+59iT2 1 + (7.65 + 7.65i)T + 59iT^{2}
61 1+(1.90−1.90i)T−61iT2 1 + (1.90 - 1.90i)T - 61iT^{2}
67 1−11.0iT−67T2 1 - 11.0iT - 67T^{2}
71 1+10.5T+71T2 1 + 10.5T + 71T^{2}
73 1+(−2.70−2.70i)T+73iT2 1 + (-2.70 - 2.70i)T + 73iT^{2}
79 1+8.32T+79T2 1 + 8.32T + 79T^{2}
83 1+9.17T+83T2 1 + 9.17T + 83T^{2}
89 1+4.25T+89T2 1 + 4.25T + 89T^{2}
97 1+(7.15+7.15i)T+97iT2 1 + (7.15 + 7.15i)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

−14.59855240063187609005930318053, −12.98025049548092383137913306227, −12.37901667288690164834641086643, −11.39500172540144681204082650881, −10.42162312147607482937588967410, −8.358868483545534764774981994394, −7.42298903810641290860573678287, −5.53037620697632889151644035045, −4.79443285457722772839826610075, −2.84000252646329249803409146391, 3.06187346871391095084032398811, 4.54159301100008861327880128511, 5.93215966959188736103960503054, 7.32166178882836081414107368594, 8.357535148674643010251516203813, 10.71534314597435587853256524963, 11.26328412528775299033848077462, 12.07366657820142949147814534885, 13.69297980386297648504775100503, 14.15685738139674212737010779481

Graph of the ZZ-function along the critical line