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Properties

Label 2-192-3.2-c2-0-11
Degree 22
Conductor 192192
Sign −0.333+0.942i-0.333 + 0.942i
Analytic cond. 5.231625.23162
Root an. cond. 2.287272.28727
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 2.82i)3-s − 5.65i·5-s + 6·7-s + (−7.00 − 5.65i)9-s + 5.65i·11-s − 10·13-s + (−16.0 − 5.65i)15-s − 22.6i·17-s + 2·19-s + (6 − 16.9i)21-s + 11.3i·23-s − 7.00·25-s + (−23.0 + 14.1i)27-s − 16.9i·29-s + 22·31-s + ⋯
L(s)  = 1  + (0.333 − 0.942i)3-s − 1.13i·5-s + 0.857·7-s + (−0.777 − 0.628i)9-s + 0.514i·11-s − 0.769·13-s + (−1.06 − 0.377i)15-s − 1.33i·17-s + 0.105·19-s + (0.285 − 0.808i)21-s + 0.491i·23-s − 0.280·25-s + (−0.851 + 0.523i)27-s − 0.585i·29-s + 0.709·31-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: −0.333+0.942i-0.333 + 0.942i
Analytic conductor: 5.231625.23162
Root analytic conductor: 2.287272.28727
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ192(65,⋅)\chi_{192} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 192, ( :1), −0.333+0.942i)(2,\ 192,\ (\ :1),\ -0.333 + 0.942i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.947179−1.33951i0.947179 - 1.33951i
L(12)L(\frac12) ≈\approx 0.947179−1.33951i0.947179 - 1.33951i
L(2)L(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+2.82i)T 1 + (-1 + 2.82i)T
good5 1+5.65iT−25T2 1 + 5.65iT - 25T^{2}
7 1−6T+49T2 1 - 6T + 49T^{2}
11 1−5.65iT−121T2 1 - 5.65iT - 121T^{2}
13 1+10T+169T2 1 + 10T + 169T^{2}
17 1+22.6iT−289T2 1 + 22.6iT - 289T^{2}
19 1−2T+361T2 1 - 2T + 361T^{2}
23 1−11.3iT−529T2 1 - 11.3iT - 529T^{2}
29 1+16.9iT−841T2 1 + 16.9iT - 841T^{2}
31 1−22T+961T2 1 - 22T + 961T^{2}
37 1−6T+1.36e3T2 1 - 6T + 1.36e3T^{2}
41 1−33.9iT−1.68e3T2 1 - 33.9iT - 1.68e3T^{2}
43 1−82T+1.84e3T2 1 - 82T + 1.84e3T^{2}
47 1+67.8iT−2.20e3T2 1 + 67.8iT - 2.20e3T^{2}
53 1−62.2iT−2.80e3T2 1 - 62.2iT - 2.80e3T^{2}
59 1−73.5iT−3.48e3T2 1 - 73.5iT - 3.48e3T^{2}
61 1−86T+3.72e3T2 1 - 86T + 3.72e3T^{2}
67 1−2T+4.48e3T2 1 - 2T + 4.48e3T^{2}
71 1−124.iT−5.04e3T2 1 - 124. iT - 5.04e3T^{2}
73 1−82T+5.32e3T2 1 - 82T + 5.32e3T^{2}
79 1+10T+6.24e3T2 1 + 10T + 6.24e3T^{2}
83 1+73.5iT−6.88e3T2 1 + 73.5iT - 6.88e3T^{2}
89 1+33.9iT−7.92e3T2 1 + 33.9iT - 7.92e3T^{2}
97 1+94T+9.40e3T2 1 + 94T + 9.40e3T^{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

−12.07533194086664122703912395565, −11.46749126503602838245904584622, −9.751796755197434385074848354339, −8.860349239253088693888270816133, −7.895983195278810667827954871521, −7.11015658890692328733828258675, −5.50803362391592836641157649931, −4.51932106171602055780806225932, −2.44784232092364310126463474209, −0.979802016761469266475686427400, 2.44172152162155152725027068676, 3.71510029959258957485593944174, 4.97674300282699422541812807906, 6.28255053067563499979673230672, 7.68884831408676266610086175556, 8.575045983740774084667128233337, 9.807868415336828821423410667152, 10.77243614813820501516033385695, 11.15273418806810480033068710746, 12.52270550173641442467250105425

Graph of the ZZ-function along the critical line