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Properties

Label 2-1280-1.1-c1-0-7
Degree 22
Conductor 12801280
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.16·3-s + 5-s + 3.16·7-s + 7.00·9-s + 6·13-s − 3.16·15-s − 2·17-s − 6.32·19-s − 10.0·21-s + 3.16·23-s + 25-s − 12.6·27-s − 4·29-s + 6.32·31-s + 3.16·35-s + 2·37-s − 18.9·39-s + 3.16·43-s + 7.00·45-s − 9.48·47-s + 3.00·49-s + 6.32·51-s + 6·53-s + 20.0·57-s + 6.32·59-s + 2·61-s + 22.1·63-s + ⋯
L(s)  = 1  − 1.82·3-s + 0.447·5-s + 1.19·7-s + 2.33·9-s + 1.66·13-s − 0.816·15-s − 0.485·17-s − 1.45·19-s − 2.18·21-s + 0.659·23-s + 0.200·25-s − 2.43·27-s − 0.742·29-s + 1.13·31-s + 0.534·35-s + 0.328·37-s − 3.03·39-s + 0.482·43-s + 1.04·45-s − 1.38·47-s + 0.428·49-s + 0.885·51-s + 0.824·53-s + 2.64·57-s + 0.823·59-s + 0.256·61-s + 2.78·63-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: 11
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: 00
Selberg data: (2, 1280, ( :1/2), 1)(2,\ 1280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1857423391.185742339
L(12)L(\frac12) ≈\approx 1.1857423391.185742339
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+3.16T+3T2 1 + 3.16T + 3T^{2}
7 1−3.16T+7T2 1 - 3.16T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−6T+13T2 1 - 6T + 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1+6.32T+19T2 1 + 6.32T + 19T^{2}
23 1−3.16T+23T2 1 - 3.16T + 23T^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1−6.32T+31T2 1 - 6.32T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1−3.16T+43T2 1 - 3.16T + 43T^{2}
47 1+9.48T+47T2 1 + 9.48T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1−6.32T+59T2 1 - 6.32T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−9.48T+67T2 1 - 9.48T + 67T^{2}
71 1−6.32T+71T2 1 - 6.32T + 71T^{2}
73 1−14T+73T2 1 - 14T + 73T^{2}
79 1+12.6T+79T2 1 + 12.6T + 79T^{2}
83 1−3.16T+83T2 1 - 3.16T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1−2T+97T2 1 - 2T + 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

−9.993989183966640743854798695329, −8.806452491469126879548474285117, −8.063211486843389197476564753252, −6.80508645682960614151316738021, −6.29472436855216926146874384811, −5.51294510289470159195864801378, −4.75069261522851296956854782525, −3.98301113428580727168680938402, −1.94604474015196483810973752484, −0.939645492925122803991613819594, 0.939645492925122803991613819594, 1.94604474015196483810973752484, 3.98301113428580727168680938402, 4.75069261522851296956854782525, 5.51294510289470159195864801378, 6.29472436855216926146874384811, 6.80508645682960614151316738021, 8.063211486843389197476564753252, 8.806452491469126879548474285117, 9.993989183966640743854798695329

Graph of the ZZ-function along the critical line