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Properties

Label 2-1280-40.27-c1-0-39
Degree 22
Conductor 12801280
Sign 0.229+0.973i0.229 + 0.973i
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  + (1.73 − 1.73i)3-s + (2 − i)5-s + (1.73 − 1.73i)7-s − 2.99i·9-s + 3.46·11-s + (−1 − i)13-s + (1.73 − 5.19i)15-s + (1 + i)17-s + 6.92i·19-s − 5.99i·21-s + (−1.73 − 1.73i)23-s + (3 − 4i)25-s − 4·29-s + 3.46i·31-s + (5.99 − 5.99i)33-s + ⋯
L(s)  = 1  + (0.999 − 0.999i)3-s + (0.894 − 0.447i)5-s + (0.654 − 0.654i)7-s − 0.999i·9-s + 1.04·11-s + (−0.277 − 0.277i)13-s + (0.447 − 1.34i)15-s + (0.242 + 0.242i)17-s + 1.58i·19-s − 1.30i·21-s + (−0.361 − 0.361i)23-s + (0.600 − 0.800i)25-s − 0.742·29-s + 0.622i·31-s + (1.04 − 1.04i)33-s + ⋯

Functional equation

Λ(s)=(1280s/2 ΓC(s) L(s)=((0.229+0.973i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1280s/2 ΓC(s+1/2) L(s)=((0.229+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12801280    =    28⋅52^{8} \cdot 5
Sign: 0.229+0.973i0.229 + 0.973i
Analytic conductor: 10.220810.2208
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1280(127,⋅)\chi_{1280} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1280, ( :1/2), 0.229+0.973i)(2,\ 1280,\ (\ :1/2),\ 0.229 + 0.973i)

Particular Values

L(1)L(1) ≈\approx 3.0360961493.036096149
L(12)L(\frac12) ≈\approx 3.0360961493.036096149
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+(−2+i)T 1 + (-2 + i)T
good3 1+(−1.73+1.73i)T−3iT2 1 + (-1.73 + 1.73i)T - 3iT^{2}
7 1+(−1.73+1.73i)T−7iT2 1 + (-1.73 + 1.73i)T - 7iT^{2}
11 1−3.46T+11T2 1 - 3.46T + 11T^{2}
13 1+(1+i)T+13iT2 1 + (1 + i)T + 13iT^{2}
17 1+(−1−i)T+17iT2 1 + (-1 - i)T + 17iT^{2}
19 1−6.92iT−19T2 1 - 6.92iT - 19T^{2}
23 1+(1.73+1.73i)T+23iT2 1 + (1.73 + 1.73i)T + 23iT^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1−3.46iT−31T2 1 - 3.46iT - 31T^{2}
37 1+(5−5i)T−37iT2 1 + (5 - 5i)T - 37iT^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+(1.73−1.73i)T−43iT2 1 + (1.73 - 1.73i)T - 43iT^{2}
47 1+(1.73−1.73i)T−47iT2 1 + (1.73 - 1.73i)T - 47iT^{2}
53 1+(7+7i)T+53iT2 1 + (7 + 7i)T + 53iT^{2}
59 1−6.92iT−59T2 1 - 6.92iT - 59T^{2}
61 1+6iT−61T2 1 + 6iT - 61T^{2}
67 1+(5.19+5.19i)T+67iT2 1 + (5.19 + 5.19i)T + 67iT^{2}
71 1−10.3iT−71T2 1 - 10.3iT - 71T^{2}
73 1+(−7+7i)T−73iT2 1 + (-7 + 7i)T - 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1+(12.1−12.1i)T−83iT2 1 + (12.1 - 12.1i)T - 83iT^{2}
89 1−8iT−89T2 1 - 8iT - 89T^{2}
97 1+(7+7i)T+97iT2 1 + (7 + 7i)T + 97iT^{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

−9.435012313204121092761444810891, −8.389767983189687045933286466531, −8.109927605313039924114717107155, −7.10036458298342385290836432137, −6.36958317014230045583027012553, −5.36384834673605009816578433895, −4.21059687140627401186053286543, −3.15169424762984126071456811197, −1.75791555210869630123897030211, −1.39638625675540828333485341571, 1.82968493364292697137694916842, 2.69345042220394502671205957342, 3.67030678293651480734874533900, 4.67693418926345662381358637912, 5.48973860372372011744900298779, 6.55066560635412489296890055329, 7.45793580869095546246580441300, 8.607182397100632813797407148349, 9.203860668932615252031428695384, 9.526625433395182113369164723701

Graph of the ZZ-function along the critical line