Properties

Label 2-192-4.3-c8-0-20
Degree $2$
Conductor $192$
Sign $1$
Analytic cond. $78.2166$
Root an. cond. $8.84402$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 46.7i·3-s + 611.·5-s − 4.42e3i·7-s − 2.18e3·9-s + 1.67e4i·11-s − 6.97e3·13-s + 2.85e4i·15-s + 4.53e4·17-s + 1.14e5i·19-s + 2.07e5·21-s + 2.72e5i·23-s − 1.68e4·25-s − 1.02e5i·27-s + 1.07e6·29-s − 1.28e6i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.978·5-s − 1.84i·7-s − 0.333·9-s + 1.14i·11-s − 0.244·13-s + 0.564i·15-s + 0.543·17-s + 0.877i·19-s + 1.06·21-s + 0.974i·23-s − 0.0431·25-s − 0.192i·27-s + 1.51·29-s − 1.39i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $1$
Analytic conductor: \(78.2166\)
Root analytic conductor: \(8.84402\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.650734240\)
\(L(\frac12)\) \(\approx\) \(2.650734240\)
\(L(5)\) 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 \( 1 - 46.7iT \)
good5 \( 1 - 611.T + 3.90e5T^{2} \)
7 \( 1 + 4.42e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.67e4iT - 2.14e8T^{2} \)
13 \( 1 + 6.97e3T + 8.15e8T^{2} \)
17 \( 1 - 4.53e4T + 6.97e9T^{2} \)
19 \( 1 - 1.14e5iT - 1.69e10T^{2} \)
23 \( 1 - 2.72e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.07e6T + 5.00e11T^{2} \)
31 \( 1 + 1.28e6iT - 8.52e11T^{2} \)
37 \( 1 + 1.10e6T + 3.51e12T^{2} \)
41 \( 1 - 4.24e6T + 7.98e12T^{2} \)
43 \( 1 + 4.82e6iT - 1.16e13T^{2} \)
47 \( 1 - 3.65e6iT - 2.38e13T^{2} \)
53 \( 1 - 9.22e6T + 6.22e13T^{2} \)
59 \( 1 - 2.35e6iT - 1.46e14T^{2} \)
61 \( 1 - 2.26e7T + 1.91e14T^{2} \)
67 \( 1 + 2.71e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.65e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.78e7T + 8.06e14T^{2} \)
79 \( 1 + 5.14e7iT - 1.51e15T^{2} \)
83 \( 1 + 1.16e7iT - 2.25e15T^{2} \)
89 \( 1 - 9.57e6T + 3.93e15T^{2} \)
97 \( 1 - 6.47e7T + 7.83e15T^{2} \)
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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

−10.66736410811257365223870145527, −10.02611627592727634557373474007, −9.551649640055803758850958330429, −7.87884453089439538572390259017, −7.03027150884772079319137301451, −5.76818918468070374262583478630, −4.55593854812008120657368591100, −3.67540111886105724888433019304, −2.03730600006449100764776744880, −0.801983461947376715639497624926, 0.888477961615842468564211405464, 2.27544486835225925276346033604, 2.91771727390497315196322424509, 5.12447450718879841108262280576, 5.87450575772689604597132787423, 6.66671521235813793907988059437, 8.366637019117366629167827237469, 8.856767837255565823139709468135, 9.951711250081728418651799502971, 11.21990485527533663919083031621

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