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Properties

Label 2-80-1.1-c3-0-4
Degree 22
Conductor 8080
Sign −1-1
Analytic cond. 4.720154.72015
Root an. cond. 2.172592.17259
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 5·5-s − 16·7-s − 11·9-s − 36·11-s − 42·13-s − 20·15-s − 110·17-s + 116·19-s + 64·21-s − 16·23-s + 25·25-s + 152·27-s + 198·29-s − 240·31-s + 144·33-s − 80·35-s − 258·37-s + 168·39-s + 442·41-s + 292·43-s − 55·45-s − 392·47-s − 87·49-s + 440·51-s + 142·53-s − 180·55-s + ⋯
L(s)  = 1  − 0.769·3-s + 0.447·5-s − 0.863·7-s − 0.407·9-s − 0.986·11-s − 0.896·13-s − 0.344·15-s − 1.56·17-s + 1.40·19-s + 0.665·21-s − 0.145·23-s + 1/5·25-s + 1.08·27-s + 1.26·29-s − 1.39·31-s + 0.759·33-s − 0.386·35-s − 1.14·37-s + 0.689·39-s + 1.68·41-s + 1.03·43-s − 0.182·45-s − 1.21·47-s − 0.253·49-s + 1.20·51-s + 0.368·53-s − 0.441·55-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: −1-1
Analytic conductor: 4.720154.72015
Root analytic conductor: 2.172592.17259
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 80, ( :3/2), −1)(2,\ 80,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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−pT 1 - p T
good3 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
7 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
11 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
13 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
17 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
19 1−116T+p3T2 1 - 116 T + p^{3} T^{2}
23 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
29 1−198T+p3T2 1 - 198 T + p^{3} T^{2}
31 1+240T+p3T2 1 + 240 T + p^{3} T^{2}
37 1+258T+p3T2 1 + 258 T + p^{3} T^{2}
41 1−442T+p3T2 1 - 442 T + p^{3} T^{2}
43 1−292T+p3T2 1 - 292 T + p^{3} T^{2}
47 1+392T+p3T2 1 + 392 T + p^{3} T^{2}
53 1−142T+p3T2 1 - 142 T + p^{3} T^{2}
59 1−348T+p3T2 1 - 348 T + p^{3} T^{2}
61 1+570T+p3T2 1 + 570 T + p^{3} T^{2}
67 1+692T+p3T2 1 + 692 T + p^{3} T^{2}
71 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
73 1+134T+p3T2 1 + 134 T + p^{3} T^{2}
79 1+784T+p3T2 1 + 784 T + p^{3} T^{2}
83 1+564T+p3T2 1 + 564 T + p^{3} T^{2}
89 1−1034T+p3T2 1 - 1034 T + p^{3} T^{2}
97 1+382T+p3T2 1 + 382 T + p^{3} T^{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

−13.22538600531996881562382386705, −12.30521513184253479476220435903, −11.12011393016412624966777750987, −10.11448004665311727661237648972, −8.990843325262090310752088601388, −7.28293716104390498989943590781, −6.05299899701135205314443605801, −4.96488709320214319464905331628, −2.75461691048782432772157823045, 0, 2.75461691048782432772157823045, 4.96488709320214319464905331628, 6.05299899701135205314443605801, 7.28293716104390498989943590781, 8.990843325262090310752088601388, 10.11448004665311727661237648972, 11.12011393016412624966777750987, 12.30521513184253479476220435903, 13.22538600531996881562382386705

Graph of the ZZ-function along the critical line