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Properties

Label 2-1280-20.3-c1-0-36
Degree 22
Conductor 12801280
Sign −0.850+0.525i-0.850 + 0.525i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)3-s + (1 − 2i)5-s + (2 + 2i)7-s − 5i·9-s − 4i·11-s + (−3 − 3i)13-s + (2 + 6i)15-s + (−3 + 3i)17-s − 8·21-s + (−6 + 6i)23-s + (−3 − 4i)25-s + (4 + 4i)27-s + 2i·29-s + 4i·31-s + (8 + 8i)33-s + ⋯
L(s)  = 1  + (−1.15 + 1.15i)3-s + (0.447 − 0.894i)5-s + (0.755 + 0.755i)7-s − 1.66i·9-s − 1.20i·11-s + (−0.832 − 0.832i)13-s + (0.516 + 1.54i)15-s + (−0.727 + 0.727i)17-s − 1.74·21-s + (−1.25 + 1.25i)23-s + (−0.600 − 0.800i)25-s + (0.769 + 0.769i)27-s + 0.371i·29-s + 0.718i·31-s + (1.39 + 1.39i)33-s + ⋯

Functional equation

Λ(s)=(1280s/2 ΓC(s) L(s)=((−0.850+0.525i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1280s/2 ΓC(s+1/2) L(s)=((−0.850+0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12801280    =    28⋅52^{8} \cdot 5
Sign: −0.850+0.525i-0.850 + 0.525i
Analytic conductor: 10.220810.2208
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1280(1023,⋅)\chi_{1280} (1023, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 1280, ( :1/2), −0.850+0.525i)(2,\ 1280,\ (\ :1/2),\ -0.850 + 0.525i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(−1+2i)T 1 + (-1 + 2i)T
good3 1+(2−2i)T−3iT2 1 + (2 - 2i)T - 3iT^{2}
7 1+(−2−2i)T+7iT2 1 + (-2 - 2i)T + 7iT^{2}
11 1+4iT−11T2 1 + 4iT - 11T^{2}
13 1+(3+3i)T+13iT2 1 + (3 + 3i)T + 13iT^{2}
17 1+(3−3i)T−17iT2 1 + (3 - 3i)T - 17iT^{2}
19 1+19T2 1 + 19T^{2}
23 1+(6−6i)T−23iT2 1 + (6 - 6i)T - 23iT^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1+(3−3i)T−37iT2 1 + (3 - 3i)T - 37iT^{2}
41 1+41T2 1 + 41T^{2}
43 1+(6−6i)T−43iT2 1 + (6 - 6i)T - 43iT^{2}
47 1+(6+6i)T+47iT2 1 + (6 + 6i)T + 47iT^{2}
53 1+(3+3i)T+53iT2 1 + (3 + 3i)T + 53iT^{2}
59 1+8T+59T2 1 + 8T + 59T^{2}
61 1+6T+61T2 1 + 6T + 61T^{2}
67 1+(−6−6i)T+67iT2 1 + (-6 - 6i)T + 67iT^{2}
71 1+12iT−71T2 1 + 12iT - 71T^{2}
73 1+(−5−5i)T+73iT2 1 + (-5 - 5i)T + 73iT^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+(6−6i)T−83iT2 1 + (6 - 6i)T - 83iT^{2}
89 1−89T2 1 - 89T^{2}
97 1+(−11+11i)T−97iT2 1 + (-11 + 11i)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

−9.510068305118607769667969282961, −8.581502404159138738954688101602, −8.060088137276290538705919340885, −6.40657561447329141454833232388, −5.61326405579647884905062035838, −5.24383173031467372298135042947, −4.49158944607474995317860323441, −3.33950870353528692256487602720, −1.68375498005259898135911888355, 0, 1.74207533632421601016936417025, 2.33522688916145357185460140694, 4.30475816928882914416401626860, 4.96278191408211701351926304571, 6.14587873090926796891878724028, 6.78014348839458342586090814749, 7.30227835306332592970253743319, 7.908996560457538008532018614127, 9.408887756577383664907424207566

Graph of the ZZ-function along the critical line