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Properties

Label 2-3888-12.11-c1-0-21
Degree 22
Conductor 38883888
Sign 0.866−0.5i0.866 - 0.5i
Analytic cond. 31.045831.0458
Root an. cond. 5.571875.57187
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  − 1.73i·7-s − 5·13-s + 5.19i·19-s + 5·25-s − 1.73i·31-s − 37-s + 12.1i·43-s + 4·49-s + 14·61-s + 3.46i·67-s + 10·73-s + 5.19i·79-s + 8.66i·91-s + 19·97-s − 3.46i·103-s + ⋯
L(s)  = 1  − 0.654i·7-s − 1.38·13-s + 1.19i·19-s + 25-s − 0.311i·31-s − 0.164·37-s + 1.84i·43-s + 0.571·49-s + 1.79·61-s + 0.423i·67-s + 1.17·73-s + 0.584i·79-s + 0.907i·91-s + 1.92·97-s − 0.341i·103-s + ⋯

Functional equation

Λ(s)=(3888s/2 ΓC(s) L(s)=((0.866−0.5i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3888s/2 ΓC(s+1/2) L(s)=((0.866−0.5i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38883888    =    24⋅352^{4} \cdot 3^{5}
Sign: 0.866−0.5i0.866 - 0.5i
Analytic conductor: 31.045831.0458
Root analytic conductor: 5.571875.57187
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3888(3887,⋅)\chi_{3888} (3887, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3888, ( :1/2), 0.866−0.5i)(2,\ 3888,\ (\ :1/2),\ 0.866 - 0.5i)

Particular Values

L(1)L(1) ≈\approx 1.5366130771.536613077
L(12)L(\frac12) ≈\approx 1.5366130771.536613077
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
good5 1−5T2 1 - 5T^{2}
7 1+1.73iT−7T2 1 + 1.73iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+5T+13T2 1 + 5T + 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1−5.19iT−19T2 1 - 5.19iT - 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+1.73iT−31T2 1 + 1.73iT - 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1−12.1iT−43T2 1 - 12.1iT - 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−14T+61T2 1 - 14T + 61T^{2}
67 1−3.46iT−67T2 1 - 3.46iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1−5.19iT−79T2 1 - 5.19iT - 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−89T2 1 - 89T^{2}
97 1−19T+97T2 1 - 19T + 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

−8.416963170923571541082869928268, −7.75518421533519040946674359356, −7.13668568638155373234807499260, −6.42733369651222158320728151636, −5.48699053123705789383172323760, −4.74115199057845646833395970396, −3.98941359928894195860114952354, −3.04854431429324880386763263848, −2.08926203324091003929469616784, −0.873410222619377677133362177247, 0.56888979240443523485451004707, 2.12866043508923671343335719466, 2.71409612348653969035337592244, 3.75224571957846151204971727794, 4.95142969122764262924888752986, 5.14471324328776300412958484157, 6.25723759574105145418968551534, 7.04533586242340999746666343824, 7.52575193544606295970146718479, 8.665539042875274873056750348447

Graph of the ZZ-function along the critical line