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Properties

Label 2-1280-40.29-c1-0-23
Degree 22
Conductor 12801280
Sign 0.856−0.515i0.856 - 0.515i
Analytic cond. 10.220810.2208
Root an. cond. 3.197003.19700
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.70·3-s + (0.539 − 2.17i)5-s + 2.63i·7-s − 0.0783·9-s + 5.41i·11-s + 6.34·13-s + (0.921 − 3.70i)15-s + 3.41i·17-s + 3.26i·19-s + 4.49i·21-s − 1.36i·23-s + (−4.41 − 2.34i)25-s − 5.26·27-s − 2i·29-s + 4.68·31-s + ⋯
L(s)  = 1  + 0.986·3-s + (0.241 − 0.970i)5-s + 0.994i·7-s − 0.0261·9-s + 1.63i·11-s + 1.75·13-s + (0.237 − 0.957i)15-s + 0.829i·17-s + 0.748i·19-s + 0.981i·21-s − 0.285i·23-s + (−0.883 − 0.468i)25-s − 1.01·27-s − 0.371i·29-s + 0.840·31-s + ⋯

Functional equation

Λ(s)=(1280s/2 ΓC(s) L(s)=((0.856−0.515i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1280s/2 ΓC(s+1/2) L(s)=((0.856−0.515i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12801280    =    28⋅52^{8} \cdot 5
Sign: 0.856−0.515i0.856 - 0.515i
Analytic conductor: 10.220810.2208
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1280(129,⋅)\chi_{1280} (129, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1280, ( :1/2), 0.856−0.515i)(2,\ 1280,\ (\ :1/2),\ 0.856 - 0.515i)

Particular Values

L(1)L(1) ≈\approx 2.4488050612.448805061
L(12)L(\frac12) ≈\approx 2.4488050612.448805061
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+(−0.539+2.17i)T 1 + (-0.539 + 2.17i)T
good3 1−1.70T+3T2 1 - 1.70T + 3T^{2}
7 1−2.63iT−7T2 1 - 2.63iT - 7T^{2}
11 1−5.41iT−11T2 1 - 5.41iT - 11T^{2}
13 1−6.34T+13T2 1 - 6.34T + 13T^{2}
17 1−3.41iT−17T2 1 - 3.41iT - 17T^{2}
19 1−3.26iT−19T2 1 - 3.26iT - 19T^{2}
23 1+1.36iT−23T2 1 + 1.36iT - 23T^{2}
29 1+2iT−29T2 1 + 2iT - 29T^{2}
31 1−4.68T+31T2 1 - 4.68T + 31T^{2}
37 1+5.75T+37T2 1 + 5.75T + 37T^{2}
41 1−7.75T+41T2 1 - 7.75T + 41T^{2}
43 1−4.44T+43T2 1 - 4.44T + 43T^{2}
47 1+4.78iT−47T2 1 + 4.78iT - 47T^{2}
53 1+1.65T+53T2 1 + 1.65T + 53T^{2}
59 1−3.26iT−59T2 1 - 3.26iT - 59T^{2}
61 1+2.49iT−61T2 1 + 2.49iT - 61T^{2}
67 1−7.86T+67T2 1 - 7.86T + 67T^{2}
71 1+6.15T+71T2 1 + 6.15T + 71T^{2}
73 1+13.5iT−73T2 1 + 13.5iT - 73T^{2}
79 1−12.6T+79T2 1 - 12.6T + 79T^{2}
83 1+14.9T+83T2 1 + 14.9T + 83T^{2}
89 1+8.52T+89T2 1 + 8.52T + 89T^{2}
97 1+4.58iT−97T2 1 + 4.58iT - 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

−9.516768034929246182854735691707, −8.728906890519987449682233534604, −8.454757129114537129131160554262, −7.60590078738788651941234883667, −6.23300564415862171317710722028, −5.64179141399862291234752299759, −4.47388609697318311892120893156, −3.65647529477294157265530110032, −2.33184866609757018041268796582, −1.57525610902437685051853742601, 0.998787871345519049870340245082, 2.66505672829299754297055220035, 3.33981540516744409999881298230, 3.99017679348804031028670187841, 5.60532697126220148590796704965, 6.36271506798405530350788427430, 7.21278864765205398535370438048, 8.073152264212579658932248489576, 8.742080818906847600076459821911, 9.448981773807819261551257181556

Graph of the ZZ-function along the critical line