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Properties

Label 2-192-12.11-c3-0-15
Degree $2$
Conductor $192$
Sign $0.942 + 0.333i$
Analytic cond. $11.3283$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.942 + 0.333i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ 0.942 + 0.333i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.51785 - 0.431995i\)
\(L(\frac12)\) \(\approx\) \(2.51785 - 0.431995i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-4.89 - 1.73i)T \)
good5 \( 1 + 16.9iT - 125T^{2} \)
7 \( 1 - 17.3iT - 343T^{2} \)
11 \( 1 - 29.3T + 1.33e3T^{2} \)
13 \( 1 - 26T + 2.19e3T^{2} \)
17 \( 1 + 67.8iT - 4.91e3T^{2} \)
19 \( 1 + 107. iT - 6.85e3T^{2} \)
23 \( 1 - 176.T + 1.21e4T^{2} \)
29 \( 1 - 16.9iT - 2.43e4T^{2} \)
31 \( 1 - 31.1iT - 2.97e4T^{2} \)
37 \( 1 + 206T + 5.06e4T^{2} \)
41 \( 1 - 305. iT - 6.89e4T^{2} \)
43 \( 1 + 93.5iT - 7.95e4T^{2} \)
47 \( 1 + 117.T + 1.03e5T^{2} \)
53 \( 1 - 50.9iT - 1.48e5T^{2} \)
59 \( 1 + 558.T + 2.05e5T^{2} \)
61 \( 1 + 278T + 2.26e5T^{2} \)
67 \( 1 - 890. iT - 3.00e5T^{2} \)
71 \( 1 - 58.7T + 3.57e5T^{2} \)
73 \( 1 + 422T + 3.89e5T^{2} \)
79 \( 1 - 668. iT - 4.93e5T^{2} \)
83 \( 1 + 29.3T + 5.71e5T^{2} \)
89 \( 1 - 373. iT - 7.04e5T^{2} \)
97 \( 1 + 1.07e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.19085426551226430899048987314, −11.15254827944322713075369402928, −9.490067613293266801697525821905, −8.995335093507615962728881915099, −8.461728742296701997767985863777, −6.99779084537303488304795442218, −5.29537595702398225095889412010, −4.48095945985781695147754493775, −2.91473677449186139773152097794, −1.26675236284137221341719895586, 1.57332790243082677000825871038, 3.23471156146629047639679896356, 3.93972410355055321193811451423, 6.28350044840629938410120434809, 7.02358482762530495570447157364, 7.898803733883064016553794049638, 9.115309503987797158300170879317, 10.32533654372711678041771153568, 10.86881274745397356498870854969, 12.20960263725676277563097833389

Graph of the $Z$-function along the critical line