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Properties

Label 2-80-1.1-c9-0-17
Degree 22
Conductor 8080
Sign −1-1
Analytic cond. 41.202841.2028
Root an. cond. 6.418946.41894
Motivic weight 99
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 204·3-s + 625·5-s − 5.43e3·7-s + 2.19e4·9-s − 7.39e4·11-s − 1.14e5·13-s + 1.27e5·15-s + 4.16e4·17-s − 1.05e6·19-s − 1.10e6·21-s − 1.59e6·23-s + 3.90e5·25-s + 4.59e5·27-s + 2.18e6·29-s + 9.61e6·31-s − 1.50e7·33-s − 3.39e6·35-s + 4.79e6·37-s − 2.33e7·39-s + 9.53e6·41-s + 1.34e7·43-s + 1.37e7·45-s − 1.14e7·47-s − 1.08e7·49-s + 8.50e6·51-s + 5.36e7·53-s − 4.62e7·55-s + ⋯
L(s)  = 1  + 1.45·3-s + 0.447·5-s − 0.855·7-s + 1.11·9-s − 1.52·11-s − 1.11·13-s + 0.650·15-s + 0.121·17-s − 1.86·19-s − 1.24·21-s − 1.19·23-s + 1/5·25-s + 0.166·27-s + 0.573·29-s + 1.87·31-s − 2.21·33-s − 0.382·35-s + 0.421·37-s − 1.61·39-s + 0.526·41-s + 0.600·43-s + 0.498·45-s − 0.342·47-s − 0.268·49-s + 0.176·51-s + 0.933·53-s − 0.680·55-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: −1-1
Analytic conductor: 41.202841.2028
Root analytic conductor: 6.418946.41894
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 80, ( :9/2), −1)(2,\ 80,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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−p4T 1 - p^{4} T
good3 1−68pT+p9T2 1 - 68 p T + p^{9} T^{2}
7 1+776pT+p9T2 1 + 776 p T + p^{9} T^{2}
11 1+73932T+p9T2 1 + 73932 T + p^{9} T^{2}
13 1+114514T+p9T2 1 + 114514 T + p^{9} T^{2}
17 1−41682T+p9T2 1 - 41682 T + p^{9} T^{2}
19 1+1057460T+p9T2 1 + 1057460 T + p^{9} T^{2}
23 1+1599336T+p9T2 1 + 1599336 T + p^{9} T^{2}
29 1−2184510T+p9T2 1 - 2184510 T + p^{9} T^{2}
31 1−9619648T+p9T2 1 - 9619648 T + p^{9} T^{2}
37 1−4799942T+p9T2 1 - 4799942 T + p^{9} T^{2}
41 1−9531882T+p9T2 1 - 9531882 T + p^{9} T^{2}
43 1−13464484T+p9T2 1 - 13464484 T + p^{9} T^{2}
47 1+11441952T+p9T2 1 + 11441952 T + p^{9} T^{2}
53 1−53615766T+p9T2 1 - 53615766 T + p^{9} T^{2}
59 1+81862620T+p9T2 1 + 81862620 T + p^{9} T^{2}
61 1+104691298T+p9T2 1 + 104691298 T + p^{9} T^{2}
67 1+2098076pT+p9T2 1 + 2098076 p T + p^{9} T^{2}
71 1+97098792T+p9T2 1 + 97098792 T + p^{9} T^{2}
73 1−171848906T+p9T2 1 - 171848906 T + p^{9} T^{2}
79 1−117380080T+p9T2 1 - 117380080 T + p^{9} T^{2}
83 1+323637636T+p9T2 1 + 323637636 T + p^{9} T^{2}
89 1+894379110T+p9T2 1 + 894379110 T + p^{9} T^{2}
97 1−232678562T+p9T2 1 - 232678562 T + p^{9} 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

−12.42934237763799951212056984893, −10.37131710355605936260390695013, −9.763972546196520475806975697300, −8.528222667281485992275981306611, −7.65521220020599115408414540802, −6.19492378195620288775281631842, −4.46740088568152008898193492567, −2.85207143831355368773919282654, −2.26613149917669568652689170921, 0, 2.26613149917669568652689170921, 2.85207143831355368773919282654, 4.46740088568152008898193492567, 6.19492378195620288775281631842, 7.65521220020599115408414540802, 8.528222667281485992275981306611, 9.763972546196520475806975697300, 10.37131710355605936260390695013, 12.42934237763799951212056984893

Graph of the ZZ-function along the critical line