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Properties

Label 2-192-24.11-c9-0-63
Degree $2$
Conductor $192$
Sign $0.244 + 0.969i$
Analytic cond. $98.8868$
Root an. cond. $9.94418$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (68.3 + 122. i)3-s + 2.06e3·5-s − 8.58e3i·7-s + (−1.03e4 + 1.67e4i)9-s − 3.82e3i·11-s − 4.28e4i·13-s + (1.41e5 + 2.53e5i)15-s − 4.27e5i·17-s − 8.92e5·19-s + (1.05e6 − 5.87e5i)21-s + 1.78e6·23-s + 2.31e6·25-s + (−2.75e6 − 1.19e5i)27-s − 1.13e6·29-s − 8.11e6i·31-s + ⋯
L(s)  = 1  + (0.487 + 0.873i)3-s + 1.47·5-s − 1.35i·7-s + (−0.524 + 0.851i)9-s − 0.0788i·11-s − 0.415i·13-s + (0.720 + 1.29i)15-s − 1.24i·17-s − 1.57·19-s + (1.18 − 0.658i)21-s + 1.32·23-s + 1.18·25-s + (−0.999 − 0.0432i)27-s − 0.296·29-s − 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.244 + 0.969i$
Analytic conductor: \(98.8868\)
Root analytic conductor: \(9.94418\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :9/2),\ 0.244 + 0.969i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.547522916\)
\(L(\frac12)\) \(\approx\) \(2.547522916\)
\(L(\frac{11}{2})\) 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 \( 1 + (-68.3 - 122. i)T \)
good5 \( 1 - 2.06e3T + 1.95e6T^{2} \)
7 \( 1 + 8.58e3iT - 4.03e7T^{2} \)
11 \( 1 + 3.82e3iT - 2.35e9T^{2} \)
13 \( 1 + 4.28e4iT - 1.06e10T^{2} \)
17 \( 1 + 4.27e5iT - 1.18e11T^{2} \)
19 \( 1 + 8.92e5T + 3.22e11T^{2} \)
23 \( 1 - 1.78e6T + 1.80e12T^{2} \)
29 \( 1 + 1.13e6T + 1.45e13T^{2} \)
31 \( 1 + 8.11e6iT - 2.64e13T^{2} \)
37 \( 1 + 1.16e7iT - 1.29e14T^{2} \)
41 \( 1 - 4.25e6iT - 3.27e14T^{2} \)
43 \( 1 + 3.31e7T + 5.02e14T^{2} \)
47 \( 1 + 1.46e7T + 1.11e15T^{2} \)
53 \( 1 + 8.39e7T + 3.29e15T^{2} \)
59 \( 1 + 1.75e7iT - 8.66e15T^{2} \)
61 \( 1 - 1.10e8iT - 1.16e16T^{2} \)
67 \( 1 + 3.73e7T + 2.72e16T^{2} \)
71 \( 1 + 1.89e8T + 4.58e16T^{2} \)
73 \( 1 - 4.06e8T + 5.88e16T^{2} \)
79 \( 1 + 3.68e8iT - 1.19e17T^{2} \)
83 \( 1 + 6.92e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.09e9iT - 3.50e17T^{2} \)
97 \( 1 + 2.50e8T + 7.60e17T^{2} \)
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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

−10.48561851966946835447893080297, −9.734777587340712266492606703792, −8.996727192961215013300403774834, −7.68792977504965025896649263827, −6.49196629582672452319752182235, −5.26916091272277780190298710832, −4.32128597880259807579178579256, −3.03504742128072299312369994085, −1.92695666066664701981848084694, −0.44516844039111295656908159614, 1.54661189870039194466442870278, 2.04786856418433537714885058981, 3.11130805395404976787688020643, 5.08247854902311464875540958460, 6.18364955232643540039351609275, 6.66075007188481068296005160068, 8.434009727362757254249045984831, 8.871436178444485214312430902003, 9.883624922698890647064486768653, 11.11429454642610683730361966149

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