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Properties

Label 2-384-12.11-c1-0-6
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  + (−1.61 − 0.618i)3-s − 1.23i·5-s + 3.23i·7-s + (2.23 + 2.00i)9-s + 0.763·11-s + 4.47·13-s + (−0.763 + 2.00i)15-s − 6.47i·17-s − 5.23i·19-s + (2.00 − 5.23i)21-s + 6.47·23-s + 3.47·25-s + (−2.38 − 4.61i)27-s + 9.23i·29-s − 0.763i·31-s + ⋯
L(s)  = 1  + (−0.934 − 0.356i)3-s − 0.552i·5-s + 1.22i·7-s + (0.745 + 0.666i)9-s + 0.230·11-s + 1.24·13-s + (−0.197 + 0.516i)15-s − 1.56i·17-s − 1.20i·19-s + (0.436 − 1.14i)21-s + 1.34·23-s + 0.694·25-s + (−0.458 − 0.888i)27-s + 1.71i·29-s − 0.137i·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 1.07028−0.197449i1.07028 - 0.197449i
L(12)L(\frac12) ≈\approx 1.07028−0.197449i1.07028 - 0.197449i
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+(1.61+0.618i)T 1 + (1.61 + 0.618i)T
good5 1+1.23iT−5T2 1 + 1.23iT - 5T^{2}
7 1−3.23iT−7T2 1 - 3.23iT - 7T^{2}
11 1−0.763T+11T2 1 - 0.763T + 11T^{2}
13 1−4.47T+13T2 1 - 4.47T + 13T^{2}
17 1+6.47iT−17T2 1 + 6.47iT - 17T^{2}
19 1+5.23iT−19T2 1 + 5.23iT - 19T^{2}
23 1−6.47T+23T2 1 - 6.47T + 23T^{2}
29 1−9.23iT−29T2 1 - 9.23iT - 29T^{2}
31 1+0.763iT−31T2 1 + 0.763iT - 31T^{2}
37 1+0.472T+37T2 1 + 0.472T + 37T^{2}
41 1−2.47iT−41T2 1 - 2.47iT - 41T^{2}
43 1−2.76iT−43T2 1 - 2.76iT - 43T^{2}
47 1−8T+47T2 1 - 8T + 47T^{2}
53 1+1.23iT−53T2 1 + 1.23iT - 53T^{2}
59 1−3.23T+59T2 1 - 3.23T + 59T^{2}
61 1+8.47T+61T2 1 + 8.47T + 61T^{2}
67 1−3.70iT−67T2 1 - 3.70iT - 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 1−7.23T+83T2 1 - 7.23T + 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

−11.39800771162205547392413200490, −10.68854169542896677749479178726, −9.084718219930451733045288532149, −8.884997237430113978137176093514, −7.32744672378307609304812102459, −6.46217092559972383772333794662, −5.36493042952581033648915175603, −4.78007082550275247062899098846, −2.86233646484382091941252710760, −1.11252385058558417766764830712, 1.21607159322177550095062969573, 3.61708637713807937652164434514, 4.25027968734526827701723628527, 5.79526940032163107620983650896, 6.49958337079848265010822460002, 7.44684296800419373091218472860, 8.642383833127025757401369485869, 9.958742619105365954791435361309, 10.70591091554262089597029801492, 10.99513743930460302127490586551

Graph of the ZZ-function along the critical line