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Properties

Label 2-192-12.11-c3-0-20
Degree 22
Conductor 192192
Sign −0.942−0.333i-0.942 - 0.333i
Analytic cond. 11.328311.3283
Root an. cond. 3.365763.36576
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 − 1.73i)3-s − 16.9i·5-s − 17.3i·7-s + (20.9 + 16.9i)9-s − 29.3·11-s + 26·13-s + (−29.3 + 83.1i)15-s − 67.8i·17-s + 107. i·19-s + (−30 + 84.8i)21-s − 176.·23-s − 162.·25-s + (−73.4 − 119. i)27-s + 16.9i·29-s − 31.1i·31-s + ⋯
L(s)  = 1  + (−0.942 − 0.333i)3-s − 1.51i·5-s − 0.935i·7-s + (0.777 + 0.628i)9-s − 0.805·11-s + 0.554·13-s + (−0.505 + 1.43i)15-s − 0.968i·17-s + 1.29i·19-s + (−0.311 + 0.881i)21-s − 1.59·23-s − 1.30·25-s + (−0.523 − 0.851i)27-s + 0.108i·29-s − 0.180i·31-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=((−0.942−0.333i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+3/2) L(s)=((−0.942−0.333i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: −0.942−0.333i-0.942 - 0.333i
Analytic conductor: 11.328311.3283
Root analytic conductor: 3.365763.36576
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ192(191,⋅)\chi_{192} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 192, ( :3/2), −0.942−0.333i)(2,\ 192,\ (\ :3/2),\ -0.942 - 0.333i)

Particular Values

L(2)L(2) ≈\approx 0.0942736+0.549466i0.0942736 + 0.549466i
L(12)L(\frac12) ≈\approx 0.0942736+0.549466i0.0942736 + 0.549466i
L(52)L(\frac{5}{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+(4.89+1.73i)T 1 + (4.89 + 1.73i)T
good5 1+16.9iT−125T2 1 + 16.9iT - 125T^{2}
7 1+17.3iT−343T2 1 + 17.3iT - 343T^{2}
11 1+29.3T+1.33e3T2 1 + 29.3T + 1.33e3T^{2}
13 1−26T+2.19e3T2 1 - 26T + 2.19e3T^{2}
17 1+67.8iT−4.91e3T2 1 + 67.8iT - 4.91e3T^{2}
19 1−107.iT−6.85e3T2 1 - 107. iT - 6.85e3T^{2}
23 1+176.T+1.21e4T2 1 + 176.T + 1.21e4T^{2}
29 1−16.9iT−2.43e4T2 1 - 16.9iT - 2.43e4T^{2}
31 1+31.1iT−2.97e4T2 1 + 31.1iT - 2.97e4T^{2}
37 1+206T+5.06e4T2 1 + 206T + 5.06e4T^{2}
41 1−305.iT−6.89e4T2 1 - 305. iT - 6.89e4T^{2}
43 1−93.5iT−7.95e4T2 1 - 93.5iT - 7.95e4T^{2}
47 1−117.T+1.03e5T2 1 - 117.T + 1.03e5T^{2}
53 1−50.9iT−1.48e5T2 1 - 50.9iT - 1.48e5T^{2}
59 1−558.T+2.05e5T2 1 - 558.T + 2.05e5T^{2}
61 1+278T+2.26e5T2 1 + 278T + 2.26e5T^{2}
67 1+890.iT−3.00e5T2 1 + 890. iT - 3.00e5T^{2}
71 1+58.7T+3.57e5T2 1 + 58.7T + 3.57e5T^{2}
73 1+422T+3.89e5T2 1 + 422T + 3.89e5T^{2}
79 1+668.iT−4.93e5T2 1 + 668. iT - 4.93e5T^{2}
83 1−29.3T+5.71e5T2 1 - 29.3T + 5.71e5T^{2}
89 1−373.iT−7.04e5T2 1 - 373. iT - 7.04e5T^{2}
97 1+1.07e3T+9.12e5T2 1 + 1.07e3T + 9.12e5T^{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

−11.74599328131394009884232945925, −10.55598659633986751871479449524, −9.743204941133572199936213886781, −8.286059964108047490627256026812, −7.51465416148392328939777908135, −6.05543776285219732014035139823, −5.09109086617813649083046122909, −4.10167408959360952729375887294, −1.50355090523781583354615996836, −0.27973072640983627071548291840, 2.37294450124460525331379994650, 3.81693371236585325364567291328, 5.46080414388818819171037104086, 6.26134310222814292622176743740, 7.23858673239299916003238330386, 8.667010718951912131482805044013, 10.08263429154997637120619718297, 10.67180690728042434204861446892, 11.48203687794604839049218485780, 12.37251512002616778523585111187

Graph of the ZZ-function along the critical line