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Properties

Label 2-192-24.11-c7-0-23
Degree $2$
Conductor $192$
Sign $0.258 - 0.965i$
Analytic cond. $59.9779$
Root an. cond. $7.74454$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 46.7·3-s + 508i·7-s + 2.18e3·9-s + 6.11e3i·13-s + 1.65e4·19-s + 2.37e4i·21-s − 7.81e4·25-s + 1.02e5·27-s + 1.78e5i·31-s + 5.49e5i·37-s + 2.86e5i·39-s + 1.24e5·43-s + 5.65e5·49-s + 7.75e5·57-s − 2.66e5i·61-s + ⋯
L(s)  = 1  + 1.00·3-s + 0.559i·7-s + 9-s + 0.772i·13-s + 0.554·19-s + 0.559i·21-s − 25-s + 1.00·27-s + 1.07i·31-s + 1.78i·37-s + 0.772i·39-s + 0.239·43-s + 0.686·49-s + 0.554·57-s − 0.150i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(59.9779\)
Root analytic conductor: \(7.74454\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :7/2),\ 0.258 - 0.965i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.882527005\)
\(L(\frac12)\) \(\approx\) \(2.882527005\)
\(L(\frac{9}{2})\) 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 \( 1 - 46.7T \)
good5 \( 1 + 7.81e4T^{2} \)
7 \( 1 - 508iT - 8.23e5T^{2} \)
11 \( 1 - 1.94e7T^{2} \)
13 \( 1 - 6.11e3iT - 6.27e7T^{2} \)
17 \( 1 - 4.10e8T^{2} \)
19 \( 1 - 1.65e4T + 8.93e8T^{2} \)
23 \( 1 + 3.40e9T^{2} \)
29 \( 1 + 1.72e10T^{2} \)
31 \( 1 - 1.78e5iT - 2.75e10T^{2} \)
37 \( 1 - 5.49e5iT - 9.49e10T^{2} \)
41 \( 1 - 1.94e11T^{2} \)
43 \( 1 - 1.24e5T + 2.71e11T^{2} \)
47 \( 1 + 5.06e11T^{2} \)
53 \( 1 + 1.17e12T^{2} \)
59 \( 1 - 2.48e12T^{2} \)
61 \( 1 + 2.66e5iT - 3.14e12T^{2} \)
67 \( 1 + 4.90e6T + 6.06e12T^{2} \)
71 \( 1 + 9.09e12T^{2} \)
73 \( 1 - 6.27e6T + 1.10e13T^{2} \)
79 \( 1 - 8.76e6iT - 1.92e13T^{2} \)
83 \( 1 - 2.71e13T^{2} \)
89 \( 1 - 4.42e13T^{2} \)
97 \( 1 - 1.22e7T + 8.07e13T^{2} \)
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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

−11.59267118985098192159644553473, −10.22422838556317897962273006125, −9.340044413796275987388468162213, −8.539702437068156978628128533607, −7.52882616105722648267511630067, −6.42044094226569646969589524413, −4.95015684441720889141804670908, −3.70453848746412829129862506439, −2.54506272691832847303934235520, −1.39615030501008385581278207994, 0.63630482202866758524296979340, 2.06402746749549703591890244834, 3.32754669221376984300443078504, 4.31250896754926498179803578231, 5.80034623202746312740307940185, 7.29810476087031974900219175679, 7.888739676689462430309850679610, 9.072675366733566207264242758055, 9.943280845599511939666147522931, 10.85568757489104292581944308859

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