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Properties

Label 2-1792-56.27-c1-0-53
Degree 22
Conductor 17921792
Sign −0.0716+0.997i-0.0716 + 0.997i
Analytic cond. 14.309114.3091
Root an. cond. 3.782743.78274
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  − 2i·3-s + 3.46·5-s + (2 − 1.73i)7-s − 9-s − 3.46·11-s + 3.46·13-s − 6.92i·15-s − 2i·19-s + (−3.46 − 4i)21-s + 3.46i·23-s + 6.99·25-s − 4i·27-s − 6i·29-s − 8·31-s + 6.92i·33-s + ⋯
L(s)  = 1  − 1.15i·3-s + 1.54·5-s + (0.755 − 0.654i)7-s − 0.333·9-s − 1.04·11-s + 0.960·13-s − 1.78i·15-s − 0.458i·19-s + (−0.755 − 0.872i)21-s + 0.722i·23-s + 1.39·25-s − 0.769i·27-s − 1.11i·29-s − 1.43·31-s + 1.20i·33-s + ⋯

Functional equation

Λ(s)=(1792s/2 ΓC(s) L(s)=((−0.0716+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0716 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1792s/2 ΓC(s+1/2) L(s)=((−0.0716+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0716 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17921792    =    28⋅72^{8} \cdot 7
Sign: −0.0716+0.997i-0.0716 + 0.997i
Analytic conductor: 14.309114.3091
Root analytic conductor: 3.782743.78274
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1792(895,⋅)\chi_{1792} (895, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1792, ( :1/2), −0.0716+0.997i)(2,\ 1792,\ (\ :1/2),\ -0.0716 + 0.997i)

Particular Values

L(1)L(1) ≈\approx 2.5370937252.537093725
L(12)L(\frac12) ≈\approx 2.5370937252.537093725
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
7 1+(−2+1.73i)T 1 + (-2 + 1.73i)T
good3 1+2iT−3T2 1 + 2iT - 3T^{2}
5 1−3.46T+5T2 1 - 3.46T + 5T^{2}
11 1+3.46T+11T2 1 + 3.46T + 11T^{2}
13 1−3.46T+13T2 1 - 3.46T + 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1+2iT−19T2 1 + 2iT - 19T^{2}
23 1−3.46iT−23T2 1 - 3.46iT - 23T^{2}
29 1+6iT−29T2 1 + 6iT - 29T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1−6.92iT−41T2 1 - 6.92iT - 41T^{2}
43 1−10.3T+43T2 1 - 10.3T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1−6iT−59T2 1 - 6iT - 59T^{2}
61 1−3.46T+61T2 1 - 3.46T + 61T^{2}
67 1−3.46T+67T2 1 - 3.46T + 67T^{2}
71 1+3.46iT−71T2 1 + 3.46iT - 71T^{2}
73 1+6.92iT−73T2 1 + 6.92iT - 73T^{2}
79 1−3.46iT−79T2 1 - 3.46iT - 79T^{2}
83 1−6iT−83T2 1 - 6iT - 83T^{2}
89 1−6.92iT−89T2 1 - 6.92iT - 89T^{2}
97 1−13.8iT−97T2 1 - 13.8iT - 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

−9.084427067543730269907437332285, −8.045217027045244029392061587241, −7.53819140957394053979307362786, −6.69008259656637490972590677662, −5.88167872646949444854727333832, −5.31390039354101702332540263653, −4.12331770300204152270433532727, −2.62292988788677763959578900250, −1.83957233770709877289524295112, −1.01000571017339276369949623800, 1.60488561404812693188391975590, 2.50588892202341957149153976129, 3.65158592882777264273271355498, 4.77484478179335369302540123919, 5.46332733702856995997139578999, 5.84598974630436377765673290252, 7.03166766925932660887166896231, 8.215814195159878537687215181709, 8.945979307412727425448825149928, 9.427441147989221749465342198191

Graph of the ZZ-function along the critical line