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Properties

Label 2-880-1.1-c1-0-12
Degree 22
Conductor 880880
Sign 11
Analytic cond. 7.026837.02683
Root an. cond. 2.650812.65081
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.37·3-s + 5-s − 3.37·7-s + 8.37·9-s + 11-s + 2·13-s + 3.37·15-s + 1.37·17-s − 0.627·19-s − 11.3·21-s − 2.74·23-s + 25-s + 18.1·27-s + 1.37·29-s − 3.37·31-s + 3.37·33-s − 3.37·35-s + 9.37·37-s + 6.74·39-s − 11.4·41-s + 4·43-s + 8.37·45-s − 2.74·47-s + 4.37·49-s + 4.62·51-s − 4.11·53-s + 55-s + ⋯
L(s)  = 1  + 1.94·3-s + 0.447·5-s − 1.27·7-s + 2.79·9-s + 0.301·11-s + 0.554·13-s + 0.870·15-s + 0.332·17-s − 0.144·19-s − 2.48·21-s − 0.572·23-s + 0.200·25-s + 3.48·27-s + 0.254·29-s − 0.605·31-s + 0.587·33-s − 0.570·35-s + 1.54·37-s + 1.07·39-s − 1.79·41-s + 0.609·43-s + 1.24·45-s − 0.400·47-s + 0.624·49-s + 0.648·51-s − 0.565·53-s + 0.134·55-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 7.026837.02683
Root analytic conductor: 2.650812.65081
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 880, ( :1/2), 1)(2,\ 880,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0554319963.055431996
L(12)L(\frac12) ≈\approx 3.0554319963.055431996
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−T 1 - T
11 1−T 1 - T
good3 1−3.37T+3T2 1 - 3.37T + 3T^{2}
7 1+3.37T+7T2 1 + 3.37T + 7T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1−1.37T+17T2 1 - 1.37T + 17T^{2}
19 1+0.627T+19T2 1 + 0.627T + 19T^{2}
23 1+2.74T+23T2 1 + 2.74T + 23T^{2}
29 1−1.37T+29T2 1 - 1.37T + 29T^{2}
31 1+3.37T+31T2 1 + 3.37T + 31T^{2}
37 1−9.37T+37T2 1 - 9.37T + 37T^{2}
41 1+11.4T+41T2 1 + 11.4T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+2.74T+47T2 1 + 2.74T + 47T^{2}
53 1+4.11T+53T2 1 + 4.11T + 53T^{2}
59 1−2.74T+59T2 1 - 2.74T + 59T^{2}
61 1+5.37T+61T2 1 + 5.37T + 61T^{2}
67 1+8T+67T2 1 + 8T + 67T^{2}
71 1+10.1T+71T2 1 + 10.1T + 71T^{2}
73 1+15.4T+73T2 1 + 15.4T + 73T^{2}
79 1−1.25T+79T2 1 - 1.25T + 79T^{2}
83 1−2.74T+83T2 1 - 2.74T + 83T^{2}
89 1+1.37T+89T2 1 + 1.37T + 89T^{2}
97 1+12.7T+97T2 1 + 12.7T + 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

−9.850487950012927257926481345266, −9.273971700822136757062469376054, −8.621437105409727728538132941072, −7.74731770890256922204915804004, −6.85794322406348863803057283312, −6.03293434937577325166887497596, −4.38179429008167304117657808191, −3.46291824474165386622167249990, −2.80319074640273897047538451374, −1.59946110200437612641870625021, 1.59946110200437612641870625021, 2.80319074640273897047538451374, 3.46291824474165386622167249990, 4.38179429008167304117657808191, 6.03293434937577325166887497596, 6.85794322406348863803057283312, 7.74731770890256922204915804004, 8.621437105409727728538132941072, 9.273971700822136757062469376054, 9.850487950012927257926481345266

Graph of the ZZ-function along the critical line