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Properties

Label 2-1584-44.43-c1-0-20
Degree 22
Conductor 15841584
Sign 0.678+0.734i0.678 + 0.734i
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  + 0.732·5-s + 2.36·7-s + (−3.23 + 0.732i)11-s − 2.36i·13-s − 6.46i·17-s + 6.46·19-s − 4.73i·23-s − 4.46·25-s + 2i·31-s + 1.73·35-s + 7.46·37-s − 4.73i·41-s + 6.46·43-s + 6.19i·47-s − 1.39·49-s + ⋯
L(s)  = 1  + 0.327·5-s + 0.895·7-s + (−0.975 + 0.220i)11-s − 0.656i·13-s − 1.56i·17-s + 1.48·19-s − 0.986i·23-s − 0.892·25-s + 0.359i·31-s + 0.293·35-s + 1.22·37-s − 0.739i·41-s + 0.986·43-s + 0.903i·47-s − 0.198·49-s + ⋯

Functional equation

Λ(s)=(1584s/2 ΓC(s) L(s)=((0.678+0.734i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1584s/2 ΓC(s+1/2) L(s)=((0.678+0.734i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15841584    =    24⋅32⋅112^{4} \cdot 3^{2} \cdot 11
Sign: 0.678+0.734i0.678 + 0.734i
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), 0.678+0.734i)(2,\ 1584,\ (\ :1/2),\ 0.678 + 0.734i)

Particular Values

L(1)L(1) ≈\approx 1.8577960671.857796067
L(12)L(\frac12) ≈\approx 1.8577960671.857796067
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+(3.23−0.732i)T 1 + (3.23 - 0.732i)T
good5 1−0.732T+5T2 1 - 0.732T + 5T^{2}
7 1−2.36T+7T2 1 - 2.36T + 7T^{2}
13 1+2.36iT−13T2 1 + 2.36iT - 13T^{2}
17 1+6.46iT−17T2 1 + 6.46iT - 17T^{2}
19 1−6.46T+19T2 1 - 6.46T + 19T^{2}
23 1+4.73iT−23T2 1 + 4.73iT - 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1−2iT−31T2 1 - 2iT - 31T^{2}
37 1−7.46T+37T2 1 - 7.46T + 37T^{2}
41 1+4.73iT−41T2 1 + 4.73iT - 41T^{2}
43 1−6.46T+43T2 1 - 6.46T + 43T^{2}
47 1−6.19iT−47T2 1 - 6.19iT - 47T^{2}
53 1−7.26T+53T2 1 - 7.26T + 53T^{2}
59 1+2.53iT−59T2 1 + 2.53iT - 59T^{2}
61 1+15.3iT−61T2 1 + 15.3iT - 61T^{2}
67 1−10.3iT−67T2 1 - 10.3iT - 67T^{2}
71 1−7.26iT−71T2 1 - 7.26iT - 71T^{2}
73 1+4.73iT−73T2 1 + 4.73iT - 73T^{2}
79 1+2.36T+79T2 1 + 2.36T + 79T^{2}
83 1−4.73T+83T2 1 - 4.73T + 83T^{2}
89 1+10.3T+89T2 1 + 10.3T + 89T^{2}
97 1+1.46T+97T2 1 + 1.46T + 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

−9.478598673420947694368964785473, −8.396663238338436338061053641816, −7.69036189641843372576028560922, −7.15413297351646408093907666918, −5.82504162679511704520928160779, −5.20665587758625698825001106260, −4.50000055807521621915864640427, −3.07137461336176476959472328365, −2.26593259916043741528726448722, −0.788870169905196896422651729223, 1.33950757465119075105545555396, 2.33092366736308160613157996120, 3.59341633309367026680212410662, 4.55750414418656188164161851743, 5.54171515033721502476657193420, 6.03515530719143611420031591447, 7.38141100102109596065697467907, 7.85863156582412435419468724888, 8.659197328860345829194765808953, 9.603087128908774590497969888194

Graph of the ZZ-function along the critical line