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Properties

Label 2-768-12.11-c1-0-25
Degree 22
Conductor 768768
Sign −0.577+0.816i-0.577 + 0.816i
Analytic cond. 6.132516.13251
Root an. cond. 2.476392.47639
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 − 1.41i)3-s − 2.82i·5-s − 2.82i·7-s + (−1.00 − 2.82i)9-s + 2·11-s + 4·13-s + (−4.00 − 2.82i)15-s + 5.65i·17-s + 2.82i·19-s + (−4.00 − 2.82i)21-s − 8·23-s − 3.00·25-s + (−5.00 − 1.41i)27-s + 2.82i·29-s − 8.48i·31-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s − 1.26i·5-s − 1.06i·7-s + (−0.333 − 0.942i)9-s + 0.603·11-s + 1.10·13-s + (−1.03 − 0.730i)15-s + 1.37i·17-s + 0.648i·19-s + (−0.872 − 0.617i)21-s − 1.66·23-s − 0.600·25-s + (−0.962 − 0.272i)27-s + 0.525i·29-s − 1.52i·31-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=((−0.577+0.816i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+1/2) L(s)=((−0.577+0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 768768    =    28⋅32^{8} \cdot 3
Sign: −0.577+0.816i-0.577 + 0.816i
Analytic conductor: 6.132516.13251
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ768(767,⋅)\chi_{768} (767, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 768, ( :1/2), −0.577+0.816i)(2,\ 768,\ (\ :1/2),\ -0.577 + 0.816i)

Particular Values

L(1)L(1) ≈\approx 0.853101−1.64806i0.853101 - 1.64806i
L(12)L(\frac12) ≈\approx 0.853101−1.64806i0.853101 - 1.64806i
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
3 1+(−1+1.41i)T 1 + (-1 + 1.41i)T
good5 1+2.82iT−5T2 1 + 2.82iT - 5T^{2}
7 1+2.82iT−7T2 1 + 2.82iT - 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1−4T+13T2 1 - 4T + 13T^{2}
17 1−5.65iT−17T2 1 - 5.65iT - 17T^{2}
19 1−2.82iT−19T2 1 - 2.82iT - 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1−2.82iT−29T2 1 - 2.82iT - 29T^{2}
31 1+8.48iT−31T2 1 + 8.48iT - 31T^{2}
37 1−4T+37T2 1 - 4T + 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1−2.82iT−43T2 1 - 2.82iT - 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−8.48iT−53T2 1 - 8.48iT - 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1−4T+61T2 1 - 4T + 61T^{2}
67 1+14.1iT−67T2 1 + 14.1iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1−2.82iT−79T2 1 - 2.82iT - 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1−5.65iT−89T2 1 - 5.65iT - 89T^{2}
97 1+6T+97T2 1 + 6T + 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

−9.855412101229841779083687311730, −8.994596815513396479366892225998, −8.182631434351678903266495681889, −7.76900236477270828771675802500, −6.44171877092891243843215888464, −5.84569945370550628814190632473, −4.15801066384631971584738330440, −3.77015201184958142725796690470, −1.80494644618733160370409467942, −0.944281193568862877142394867146, 2.25943473043660764430411741962, 3.07660828285776428562006821508, 4.00130827318759063374247396910, 5.27042862733379677804268421892, 6.22670110820432856620595626044, 7.12465929549730367019309647976, 8.281710285438635550797649542370, 8.961369816676942957874767219788, 9.765423492759884298144469063424, 10.51790162432362088151821340791

Graph of the ZZ-function along the critical line