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Properties

Label 2-1280-1.1-c1-0-16
Degree 22
Conductor 12801280
Sign −1-1
Analytic cond. 10.220810.2208
Root an. cond. 3.197003.19700
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·3-s + 5-s − 4.73·7-s + 4.46·9-s + 3.46·11-s + 3.46·13-s − 2.73·15-s − 3.46·17-s + 2·19-s + 12.9·21-s + 2.19·23-s + 25-s − 3.99·27-s − 2.53·31-s − 9.46·33-s − 4.73·35-s − 6·37-s − 9.46·39-s − 9.46·41-s − 0.196·43-s + 4.46·45-s + 2.19·47-s + 15.3·49-s + 9.46·51-s − 10.3·53-s + 3.46·55-s − 5.46·57-s + ⋯
L(s)  = 1  − 1.57·3-s + 0.447·5-s − 1.78·7-s + 1.48·9-s + 1.04·11-s + 0.960·13-s − 0.705·15-s − 0.840·17-s + 0.458·19-s + 2.82·21-s + 0.457·23-s + 0.200·25-s − 0.769·27-s − 0.455·31-s − 1.64·33-s − 0.799·35-s − 0.986·37-s − 1.51·39-s − 1.47·41-s − 0.0299·43-s + 0.665·45-s + 0.320·47-s + 2.19·49-s + 1.32·51-s − 1.42·53-s + 0.467·55-s − 0.723·57-s + ⋯

Functional equation

Λ(s)=(1280s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1280s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12801280    =    28⋅52^{8} \cdot 5
Sign: −1-1
Analytic conductor: 10.220810.2208
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1280, ( :1/2), −1)(2,\ 1280,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
good3 1+2.73T+3T2 1 + 2.73T + 3T^{2}
7 1+4.73T+7T2 1 + 4.73T + 7T^{2}
11 1−3.46T+11T2 1 - 3.46T + 11T^{2}
13 1−3.46T+13T2 1 - 3.46T + 13T^{2}
17 1+3.46T+17T2 1 + 3.46T + 17T^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
23 1−2.19T+23T2 1 - 2.19T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1+2.53T+31T2 1 + 2.53T + 31T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1+9.46T+41T2 1 + 9.46T + 41T^{2}
43 1+0.196T+43T2 1 + 0.196T + 43T^{2}
47 1−2.19T+47T2 1 - 2.19T + 47T^{2}
53 1+10.3T+53T2 1 + 10.3T + 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1+0.928T+61T2 1 + 0.928T + 61T^{2}
67 1−0.196T+67T2 1 - 0.196T + 67T^{2}
71 1+16.3T+71T2 1 + 16.3T + 71T^{2}
73 1+6.39T+73T2 1 + 6.39T + 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1+1.26T+83T2 1 + 1.26T + 83T^{2}
89 1−12.9T+89T2 1 - 12.9T + 89T^{2}
97 1−14.3T+97T2 1 - 14.3T + 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

−9.369617944511913742317556790155, −8.788555865680087591935935879703, −7.05888379108100852942441508436, −6.57444068311977131761921957332, −6.08829428817590354920690143168, −5.28995590108713009777018455637, −4.10912843233860059316949401395, −3.17331369713556042964900992061, −1.38644316172211549871292105859, 0, 1.38644316172211549871292105859, 3.17331369713556042964900992061, 4.10912843233860059316949401395, 5.28995590108713009777018455637, 6.08829428817590354920690143168, 6.57444068311977131761921957332, 7.05888379108100852942441508436, 8.788555865680087591935935879703, 9.369617944511913742317556790155

Graph of the ZZ-function along the critical line