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Properties

Label 2-1584-44.43-c1-0-10
Degree 22
Conductor 15841584
Sign 0.996+0.0791i0.996 + 0.0791i
Analytic cond. 12.648312.6483
Root an. cond. 3.556443.55644
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  − 2.73·5-s − 5.13·7-s + (−1.88 + 2.73i)11-s − 5.13i·13-s + 3.76i·17-s + 3.76·19-s + 1.26i·23-s + 2.46·25-s − 2i·31-s + 14.0·35-s + 0.535·37-s − 10.2i·41-s + 3.76·43-s + 4.19i·47-s + 19.3·49-s + ⋯
L(s)  = 1  − 1.22·5-s − 1.94·7-s + (−0.566 + 0.823i)11-s − 1.42i·13-s + 0.912i·17-s + 0.862·19-s + 0.264i·23-s + 0.492·25-s − 0.359i·31-s + 2.37·35-s + 0.0881·37-s − 1.60i·41-s + 0.573·43-s + 0.612i·47-s + 2.77·49-s + ⋯

Functional equation

Λ(s)=(1584s/2 ΓC(s) L(s)=((0.996+0.0791i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0791i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1584s/2 ΓC(s+1/2) L(s)=((0.996+0.0791i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15841584    =    24⋅32⋅112^{4} \cdot 3^{2} \cdot 11
Sign: 0.996+0.0791i0.996 + 0.0791i
Analytic conductor: 12.648312.6483
Root analytic conductor: 3.556443.55644
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1584(703,⋅)\chi_{1584} (703, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1584, ( :1/2), 0.996+0.0791i)(2,\ 1584,\ (\ :1/2),\ 0.996 + 0.0791i)

Particular Values

L(1)L(1) ≈\approx 0.69154636620.6915463662
L(12)L(\frac12) ≈\approx 0.69154636620.6915463662
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
11 1+(1.88−2.73i)T 1 + (1.88 - 2.73i)T
good5 1+2.73T+5T2 1 + 2.73T + 5T^{2}
7 1+5.13T+7T2 1 + 5.13T + 7T^{2}
13 1+5.13iT−13T2 1 + 5.13iT - 13T^{2}
17 1−3.76iT−17T2 1 - 3.76iT - 17T^{2}
19 1−3.76T+19T2 1 - 3.76T + 19T^{2}
23 1−1.26iT−23T2 1 - 1.26iT - 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
37 1−0.535T+37T2 1 - 0.535T + 37T^{2}
41 1+10.2iT−41T2 1 + 10.2iT - 41T^{2}
43 1−3.76T+43T2 1 - 3.76T + 43T^{2}
47 1−4.19iT−47T2 1 - 4.19iT - 47T^{2}
53 1−10.7T+53T2 1 - 10.7T + 53T^{2}
59 1−9.46iT−59T2 1 - 9.46iT - 59T^{2}
61 1−2.38iT−61T2 1 - 2.38iT - 61T^{2}
67 1−10.3iT−67T2 1 - 10.3iT - 67T^{2}
71 1+10.7iT−71T2 1 + 10.7iT - 71T^{2}
73 1+10.2iT−73T2 1 + 10.2iT - 73T^{2}
79 1−5.13T+79T2 1 - 5.13T + 79T^{2}
83 1+10.2T+83T2 1 + 10.2T + 83T^{2}
89 1−10.3T+89T2 1 - 10.3T + 89T^{2}
97 1−5.46T+97T2 1 - 5.46T + 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

−9.475118209314111554973841736861, −8.598277221303195924618230929176, −7.53385070333210197779980223463, −7.32489936962704426752280907022, −6.13463901697323520316804110426, −5.42157485564327696847866596917, −4.08522439675853833574654485847, −3.45603162113302516051925250032, −2.63733891581698118175310761670, −0.53541363683619127872032832978, 0.57339477833496142559792334339, 2.69976076683283500719891765888, 3.42278131420198372643253919985, 4.16481302312180365604462608312, 5.32932665255692698316246617778, 6.41957272127299413311823283827, 6.95885846283835998094679823177, 7.75147351997956101716184493330, 8.725617524795713462793757011267, 9.426473281374356046142004029780

Graph of the ZZ-function along the critical line