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Properties

Label 2-1280-40.29-c1-0-6
Degree 22
Conductor 12801280
Sign −0.515−0.856i-0.515 - 0.856i
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)=((−0.515−0.856i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1280s/2 ΓC(s+1/2) L(s)=((−0.515−0.856i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12801280    =    28⋅52^{8} \cdot 5
Sign: −0.515−0.856i-0.515 - 0.856i
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), −0.515−0.856i)(2,\ 1280,\ (\ :1/2),\ -0.515 - 0.856i)

Particular Values

L(1)L(1) ≈\approx 1.3525913631.352591363
L(12)L(\frac12) ≈\approx 1.3525913631.352591363
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

−9.890015363650159060499460817932, −9.352621650423584344939965201940, −8.049646905229552278262828247840, −7.35220680178105519110396257887, −7.20611676137382239056844687602, −5.86781359026046147898049134561, −4.49520285817842848799570127262, −3.84928507030962824134825197254, −2.74417669207143747667385092954, −1.95454611415069399336704424213, 0.45898161306143610099975672638, 2.36669993210522031120577544069, 2.90260846034973317998639208661, 4.15742727427055257248847047664, 5.25607739062761704418742915441, 5.77665885739156199210369026857, 7.20530313822267613727686314847, 8.031080447586486175186612647438, 8.665364437768847111870982766317, 9.211460762088490026808877735936

Graph of the ZZ-function along the critical line