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Properties

Label 2-768-24.11-c1-0-17
Degree 22
Conductor 768768
Sign 0.408+0.912i0.408 + 0.912i
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  + (0.618 − 1.61i)3-s + 1.23·5-s + 3.23i·7-s + (−2.23 − 2.00i)9-s − 0.763i·11-s − 4.47i·13-s + (0.763 − 2.00i)15-s − 6.47i·17-s + 5.23·19-s + (5.23 + 2.00i)21-s + 6.47·23-s − 3.47·25-s + (−4.61 + 2.38i)27-s + 9.23·29-s + 0.763i·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s + 0.552·5-s + 1.22i·7-s + (−0.745 − 0.666i)9-s − 0.230i·11-s − 1.24i·13-s + (0.197 − 0.516i)15-s − 1.56i·17-s + 1.20·19-s + (1.14 + 0.436i)21-s + 1.34·23-s − 0.694·25-s + (−0.888 + 0.458i)27-s + 1.71·29-s + 0.137i·31-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=((0.408+0.912i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+1/2) L(s)=((0.408+0.912i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 768768    =    28⋅32^{8} \cdot 3
Sign: 0.408+0.912i0.408 + 0.912i
Analytic conductor: 6.132516.13251
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ768(383,⋅)\chi_{768} (383, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 768, ( :1/2), 0.408+0.912i)(2,\ 768,\ (\ :1/2),\ 0.408 + 0.912i)

Particular Values

L(1)L(1) ≈\approx 1.56985−1.01762i1.56985 - 1.01762i
L(12)L(\frac12) ≈\approx 1.56985−1.01762i1.56985 - 1.01762i
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+(−0.618+1.61i)T 1 + (-0.618 + 1.61i)T
good5 1−1.23T+5T2 1 - 1.23T + 5T^{2}
7 1−3.23iT−7T2 1 - 3.23iT - 7T^{2}
11 1+0.763iT−11T2 1 + 0.763iT - 11T^{2}
13 1+4.47iT−13T2 1 + 4.47iT - 13T^{2}
17 1+6.47iT−17T2 1 + 6.47iT - 17T^{2}
19 1−5.23T+19T2 1 - 5.23T + 19T^{2}
23 1−6.47T+23T2 1 - 6.47T + 23T^{2}
29 1−9.23T+29T2 1 - 9.23T + 29T^{2}
31 1−0.763iT−31T2 1 - 0.763iT - 31T^{2}
37 1+0.472iT−37T2 1 + 0.472iT - 37T^{2}
41 1+2.47iT−41T2 1 + 2.47iT - 41T^{2}
43 1−2.76T+43T2 1 - 2.76T + 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1−1.23T+53T2 1 - 1.23T + 53T^{2}
59 1+3.23iT−59T2 1 + 3.23iT - 59T^{2}
61 1−8.47iT−61T2 1 - 8.47iT - 61T^{2}
67 1+3.70T+67T2 1 + 3.70T + 67T^{2}
71 1+11.4T+71T2 1 + 11.4T + 71T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1−13.7iT−79T2 1 - 13.7iT - 79T^{2}
83 1−7.23iT−83T2 1 - 7.23iT - 83T^{2}
89 1−4iT−89T2 1 - 4iT - 89T^{2}
97 1+8.47T+97T2 1 + 8.47T + 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

−9.933578679137095816712990570155, −9.203463525811448549476932953378, −8.483212531289877669703548751814, −7.59798614506119529994196141514, −6.71368811642471717302431040962, −5.66507361989607478536249095400, −5.17121339150905459758234223314, −3.03795318017681575240638062702, −2.62124695530669829008843234732, −1.02081493097523189981406554888, 1.57529871934387436279978745277, 3.11232730619813553288894916702, 4.13399543037172595332335176138, 4.82030855904975558518792360855, 6.05194532522764385833944496954, 7.01588506882836258599405623548, 8.008908528838254671261920918595, 8.958271565234225082256350097080, 9.750953125497315800468477289560, 10.32318757223453145073239624383

Graph of the ZZ-function along the critical line