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Properties

Label 2-384-12.11-c1-0-9
Degree 22
Conductor 384384
Sign 0.934+0.356i0.934 + 0.356i
Analytic cond. 3.066253.06625
Root an. cond. 1.751071.75107
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 + 3.23i·5-s − 1.23i·7-s + (−2.23 − 2.00i)9-s + 5.23·11-s + 4.47·13-s + (5.23 + 2.00i)15-s − 2.47i·17-s + 0.763i·19-s + (−2.00 − 0.763i)21-s + 2.47·23-s − 5.47·25-s + (−4.61 + 2.38i)27-s + 4.76i·29-s − 5.23i·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s + 1.44i·5-s − 0.467i·7-s + (−0.745 − 0.666i)9-s + 1.57·11-s + 1.24·13-s + (1.35 + 0.516i)15-s − 0.599i·17-s + 0.175i·19-s + (−0.436 − 0.166i)21-s + 0.515·23-s − 1.09·25-s + (−0.888 + 0.458i)27-s + 0.884i·29-s − 0.940i·31-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.934+0.356i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((0.934+0.356i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.934+0.356i0.934 + 0.356i
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ384(383,⋅)\chi_{384} (383, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :1/2), 0.934+0.356i)(2,\ 384,\ (\ :1/2),\ 0.934 + 0.356i)

Particular Values

L(1)L(1) ≈\approx 1.60962−0.296948i1.60962 - 0.296948i
L(12)L(\frac12) ≈\approx 1.60962−0.296948i1.60962 - 0.296948i
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 1−3.23iT−5T2 1 - 3.23iT - 5T^{2}
7 1+1.23iT−7T2 1 + 1.23iT - 7T^{2}
11 1−5.23T+11T2 1 - 5.23T + 11T^{2}
13 1−4.47T+13T2 1 - 4.47T + 13T^{2}
17 1+2.47iT−17T2 1 + 2.47iT - 17T^{2}
19 1−0.763iT−19T2 1 - 0.763iT - 19T^{2}
23 1−2.47T+23T2 1 - 2.47T + 23T^{2}
29 1−4.76iT−29T2 1 - 4.76iT - 29T^{2}
31 1+5.23iT−31T2 1 + 5.23iT - 31T^{2}
37 1+8.47T+37T2 1 + 8.47T + 37T^{2}
41 1−6.47iT−41T2 1 - 6.47iT - 41T^{2}
43 1+7.23iT−43T2 1 + 7.23iT - 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1−3.23iT−53T2 1 - 3.23iT - 53T^{2}
59 1+1.23T+59T2 1 + 1.23T + 59T^{2}
61 1+0.472T+61T2 1 + 0.472T + 61T^{2}
67 1−9.70iT−67T2 1 - 9.70iT - 67T^{2}
71 1+15.4T+71T2 1 + 15.4T + 71T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1+0.291iT−79T2 1 + 0.291iT - 79T^{2}
83 1−2.76T+83T2 1 - 2.76T + 83T^{2}
89 1−4iT−89T2 1 - 4iT - 89T^{2}
97 1−0.472T+97T2 1 - 0.472T + 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

−11.35938087296978120152053947417, −10.56718514876331542746735153428, −9.342747798133658534827646705644, −8.460887599165852441627775716635, −7.18525770703991326931854786251, −6.79698648521432253349227155231, −5.93826331989077339506237106707, −3.86457554916008562357222693846, −3.03819750642008649545686305237, −1.45472451603873987349616811025, 1.51012220631758179412520442508, 3.52229369124652651480767571762, 4.38230861728745604736417430792, 5.38207263894208148420383547763, 6.40602062376300254341970292162, 8.160166407138318214300651917701, 8.968546524243489079159293704635, 9.110588519555404409207739862895, 10.38547358187732430650582116834, 11.45064716632547921795721179282

Graph of the ZZ-function along the critical line