Properties

Degree $2$
Conductor $192$
Sign $0.569 + 0.821i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (0.909 + 0.909i)5-s + 0.654·7-s + 2.99i·9-s + (13.3 − 13.3i)11-s + (8.32 − 8.32i)13-s − 2.22i·15-s − 3.93·17-s + (−16.8 − 16.8i)19-s + (−0.801 − 0.801i)21-s + 23.1·23-s − 23.3i·25-s + (3.67 − 3.67i)27-s + (35.6 − 35.6i)29-s + 45.5i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.181 + 0.181i)5-s + 0.0935·7-s + 0.333i·9-s + (1.21 − 1.21i)11-s + (0.640 − 0.640i)13-s − 0.148i·15-s − 0.231·17-s + (−0.889 − 0.889i)19-s + (−0.0381 − 0.0381i)21-s + 1.00·23-s − 0.933i·25-s + (0.136 − 0.136i)27-s + (1.22 − 1.22i)29-s + 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.569 + 0.821i$
Motivic weight: \(2\)
Character: $\chi_{192} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24998 - 0.654568i\)
\(L(\frac12)\) \(\approx\) \(1.24998 - 0.654568i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (-0.909 - 0.909i)T + 25iT^{2} \)
7 \( 1 - 0.654T + 49T^{2} \)
11 \( 1 + (-13.3 + 13.3i)T - 121iT^{2} \)
13 \( 1 + (-8.32 + 8.32i)T - 169iT^{2} \)
17 \( 1 + 3.93T + 289T^{2} \)
19 \( 1 + (16.8 + 16.8i)T + 361iT^{2} \)
23 \( 1 - 23.1T + 529T^{2} \)
29 \( 1 + (-35.6 + 35.6i)T - 841iT^{2} \)
31 \( 1 - 45.5iT - 961T^{2} \)
37 \( 1 + (-10.1 - 10.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 28.4iT - 1.68e3T^{2} \)
43 \( 1 + (22.7 - 22.7i)T - 1.84e3iT^{2} \)
47 \( 1 - 10.7iT - 2.20e3T^{2} \)
53 \( 1 + (-41.5 - 41.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (-21.0 + 21.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (68.7 - 68.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (67.8 + 67.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 33.3T + 5.04e3T^{2} \)
73 \( 1 - 18.6iT - 5.32e3T^{2} \)
79 \( 1 - 6.29iT - 6.24e3T^{2} \)
83 \( 1 + (-72.0 - 72.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 10.6iT - 7.92e3T^{2} \)
97 \( 1 - 143.T + 9.40e3T^{2} \)
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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

−12.05313428872087786830448225152, −11.20164853397477906397488954023, −10.45029309957100921357030776679, −8.976706973194079817440356422957, −8.216115269877501449593733891294, −6.64621779271117959408007360640, −6.12415670183239467039815755246, −4.59460777317616314944557983637, −2.98848440125084608347592057126, −0.994332729000749753185258906975, 1.64485216200630904756813040301, 3.82572984551057280577737066098, 4.81474735482928733396929625419, 6.21457770057744251420834720243, 7.10769655466420844712566315217, 8.711504200373937837769183283038, 9.457354163359421793246297895315, 10.50476846576218431927289618142, 11.51086201760934106886953477592, 12.32674506311227112715724143369

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