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Properties

Label 2-384-8.5-c1-0-0
Degree 22
Conductor 384384
Sign −0.707−0.707i-0.707 - 0.707i
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  + i·3-s − 4·7-s − 9-s + 4i·11-s + 4i·13-s − 2·17-s + 4i·19-s − 4i·21-s − 8·23-s + 5·25-s − i·27-s − 8i·29-s + 4·31-s − 4·33-s + 4i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.51·7-s − 0.333·9-s + 1.20i·11-s + 1.10i·13-s − 0.485·17-s + 0.917i·19-s − 0.872i·21-s − 1.66·23-s + 25-s − 0.192i·27-s − 1.48i·29-s + 0.718·31-s − 0.696·33-s + 0.657i·37-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ384(193,⋅)\chi_{384} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :1/2), −0.707−0.707i)(2,\ 384,\ (\ :1/2),\ -0.707 - 0.707i)

Particular Values

L(1)L(1) ≈\approx 0.293958+0.709678i0.293958 + 0.709678i
L(12)L(\frac12) ≈\approx 0.293958+0.709678i0.293958 + 0.709678i
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−iT 1 - iT
good5 1−5T2 1 - 5T^{2}
7 1+4T+7T2 1 + 4T + 7T^{2}
11 1−4iT−11T2 1 - 4iT - 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1+8iT−29T2 1 + 8iT - 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−4iT−37T2 1 - 4iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−8T+47T2 1 - 8T + 47T^{2}
53 1−8iT−53T2 1 - 8iT - 53T^{2}
59 1+12iT−59T2 1 + 12iT - 59T^{2}
61 1−12iT−61T2 1 - 12iT - 61T^{2}
67 1+12iT−67T2 1 + 12iT - 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−11.87155743444823576618691414129, −10.47480634983299212945983286650, −9.795043619028065311157014176521, −9.289120191022971870579262648242, −8.022973344092631875977933620594, −6.74300109421012252853259468529, −6.12222230875314374987170042074, −4.58660694737531538003861714253, −3.75572216193165549442230518239, −2.30795357271118856102465366246, 0.48679498189518907930183266173, 2.70932525750295076442157014770, 3.59263299891930224026818578911, 5.39213430206201839831945637149, 6.30498517475523438238710455379, 7.04387989340392764368635757257, 8.303111151110790689211383562790, 9.031461175966213677931290791137, 10.18122931162721848567067732934, 10.91923308779591399917835911238

Graph of the ZZ-function along the critical line