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Properties

Label 2-1280-40.3-c1-0-22
Degree 22
Conductor 12801280
Sign 0.973+0.229i0.973 + 0.229i
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)=((0.973+0.229i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1280s/2 ΓC(s+1/2) L(s)=((0.973+0.229i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12801280    =    28⋅52^{8} \cdot 5
Sign: 0.973+0.229i0.973 + 0.229i
Analytic conductor: 10.220810.2208
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1280(383,⋅)\chi_{1280} (383, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1280, ( :1/2), 0.973+0.229i)(2,\ 1280,\ (\ :1/2),\ 0.973 + 0.229i)

Particular Values

L(1)L(1) ≈\approx 1.9283303231.928330323
L(12)L(\frac12) ≈\approx 1.9283303231.928330323
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

−9.575770751682785259304893170341, −8.765852303542415923673902903029, −8.397267650988539927376904809920, −7.20924957123185824353214197929, −6.58325554751861152459748669912, −4.99218500360125379989454933050, −4.29358123229920425813289483807, −3.55991014931965785906009800304, −2.86512113557611190551058533997, −0.830057158570300643312637949553, 1.31482852011629183228314243358, 2.53742082632452432688807137161, 3.42233251412544790252163464322, 4.15252359049748957888910733822, 5.87858510084929993875931566238, 6.54924465860822829057565795998, 7.37991733968859590875168139616, 8.022924475940615697521075816568, 8.758256800088369239371849489072, 9.413022989873941316093826761235

Graph of the ZZ-function along the critical line