Properties

Label 2-192-4.3-c8-0-19
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 46.7i·3-s + 1.15e3·5-s − 1.31e3i·7-s − 2.18e3·9-s + 1.50e4i·11-s + 3.08e4·13-s − 5.42e4i·15-s + 4.64e4·17-s + 6.91e4i·19-s − 6.13e4·21-s + 3.93e5i·23-s + 9.53e5·25-s + 1.02e5i·27-s + 7.74e5·29-s + 4.66e5i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.85·5-s − 0.546i·7-s − 0.333·9-s + 1.02i·11-s + 1.08·13-s − 1.07i·15-s + 0.556·17-s + 0.530i·19-s − 0.315·21-s + 1.40i·23-s + 2.44·25-s + 0.192i·27-s + 1.09·29-s + 0.505i·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

\(L(\frac{9}{2})\) \(\approx\) \(3.493442293\)
\(L(\frac12)\) \(\approx\) \(3.493442293\)
\(L(5)\) 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 \( 1 - 1.15e3T + 3.90e5T^{2} \)
7 \( 1 + 1.31e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.50e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.08e4T + 8.15e8T^{2} \)
17 \( 1 - 4.64e4T + 6.97e9T^{2} \)
19 \( 1 - 6.91e4iT - 1.69e10T^{2} \)
23 \( 1 - 3.93e5iT - 7.83e10T^{2} \)
29 \( 1 - 7.74e5T + 5.00e11T^{2} \)
31 \( 1 - 4.66e5iT - 8.52e11T^{2} \)
37 \( 1 + 4.99e5T + 3.51e12T^{2} \)
41 \( 1 + 3.81e6T + 7.98e12T^{2} \)
43 \( 1 - 2.22e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.92e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.22e7T + 6.22e13T^{2} \)
59 \( 1 + 2.04e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.27e7T + 1.91e14T^{2} \)
67 \( 1 - 1.90e7iT - 4.06e14T^{2} \)
71 \( 1 + 4.88e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.46e7T + 8.06e14T^{2} \)
79 \( 1 - 6.22e7iT - 1.51e15T^{2} \)
83 \( 1 - 7.55e6iT - 2.25e15T^{2} \)
89 \( 1 - 7.40e7T + 3.93e15T^{2} \)
97 \( 1 - 1.23e8T + 7.83e15T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.86807791187497380371087269970, −9.995429955951100755220320899939, −9.277540475059074123632853528627, −7.965496892275030456111644289887, −6.75243942385422568895096287552, −6.00722352529113375385940666987, −4.94606781757574283794872686886, −3.20876253574529029192448701129, −1.76685030325429022050789502056, −1.25245125801738970823067313166, 0.872553512375313053431732076273, 2.21095168720114380665538858131, 3.25831760791925354112905429506, 4.95106097882065864403936060571, 5.87099702049449038414148850574, 6.47878428311398501864813527948, 8.556078934155657728124052676006, 8.998130820647177465634783971080, 10.18083966031906567347794593643, 10.70482766135435140510684672826

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